IHK 


RAILROAD    CURVES 


AND 


EARTHWORK 


BY 


•      C.   FRANK  ALLEN,  ,.B. 

MEMBER  AMERICAN  SOCIETY  OF  CIVIL   ENGINEERS 

PROFESSOR  OF  RAILROAD   ENGINEERING  IN    THE    MASSACHUSETTS 

INSTITUTE  OF  TECHNOLOGY 


RI)  ^EDITION 


NEW  YORK 
SPON  &  CHAMBERLAIN,   123  LIBERTY  STREET 

LONDON 
E.  &  F.  N.  SPON,  LTD.,  125  STRAND 

1903 


b 
/ 


COPYRIGHT,  1889,  1894,  AND 
BT  C.  F.  ALLEN. 


Nortoooto 

J.  S.  Cashing  &  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE. 


THIS  book  was  prepared  for  the  use  of  the  students  in  the 
author's  classes.  It  has  been  used  in  lithographed  sheets  for  a- 
number  of  years  in  very  nearly  the  present  form,  and  has  given 
satisfaction  sufficient  to  suggest  putting  it  in  print.  An  effort 
has  been  made  to  have  the  demonstrations  simple  and  direct, 
and  special  care  has  been  given  to  the  arrangement  and  the 
typography,  in  order  to  secure  clearness  and  conciseness  of 
mathematical  statement.  Much  of  the  material  in  the  earlier 
part  of  the  book  is  necessarily  similar  to  that  found  in  one  or 
more  of  several  excellent  field  books,  although  the  methods  of 
demonstration  are  in  many  cases  new.  This  will  be  found  true 
especially  in  Compound  Curves,  for  which  simple  treatment 
has  been  found  quite  possible.  New  material  will  be  found  in 
the  chapters  on  Turnouts  and  on  "  Y"  Tracks  and  Crossings. 
The  Spiral  Easement  Curve  is  treated  originally.  The  chapters 
on  Earthwork  are  essentially  new ;  they  include  Staking  Out ; 
Computation,  directly  and  with  Tables  and  Diagrams ;  also 
Haul,  treated  ordinarily  and  by  Mass  Diagram.  Most  of  the 
material  relating  to  Earthwork  is  not  elsewhere  readily  available 
for  students'  use. 

The  book  has  been  written  especially  to  meet  the  needs  of 
students  in  engineering  colleges,  but  it  is  probable  that  it  will 
be  found  useful  by  many  engineers  in  practice.  The  size  of 
page  allows  it  to  be  used  as  a  pocket  book  in  the  field.  It  is 
difficult  to  avoid  typographical  and  clerical  errors  ;  the  author 
will  consider  it  a  favor  if  he  is  notified  of  any  errors  found  to 
exist. 

C.  FRANK  ALLEN. 
BOSTON, 

September,  1899. 

in 


PREFACE   TO   THIRD   EDITION. 


IN  this  edition  the  chapter  on  the  Spiral  Easement  Curve 
has  been  entirely  re-written  and  enlarged ;  the  treatment  is 
largely  new  as  to  detail.  The  author  desires,  however,  to  ac- 
knowledge his  indebtedness  to  the  paper  on  Transition  Curves, 
by  W.  B.  Lee,  in  the  Transactions  of  the  American  Society  of 
Civil  Engineers,  December,  1901.  Two  pages  have  been  added 
to  the  chapter  on  Special  Problems  in  Earthwork.  The  refer- 
ences to  Tables  have  been  changed  so  as  to  apply  to  the  newly 
published  Tables  by  the  author  of  this  book. 

C.   FRANK  ALLEN. 
July,  1903. 


iv 


CONTENTS. 


CHAPTER  I. 

RECONNOISSANCE. 

SECTION  TAGE 

1.  Operations  in  location 1 

2.  Reconnoissance 1 

3.  Nature  of  examination 1 

4.  Features  of  topography 2 

5.  Purposes  of  reconnoissance 3 

6.  Elevations,  how  taken 3 

7.  Pocket  instruments  used 4 

8.  Importance  of  reconnoissance 6 

CHAPTER  II. 
PRELIMINARY  SURVEY. 

9.  Nature  of  preliminary 6 

10.  Grades 6 

11.  Importance  of  low  grades 7 

12.  Pusher  grades 7 

13-14.  Purposes  of  preliminary 8 

15.  Nature 9 

16.  Methods 9 

17.  Backing  up 10 

18.  Notes 11 

19.  Organization  of  party 11 

20.  Locating  engineer 11 

21.  Transitman  ;  also  form  of  notes 12 

22.  Head  chainman 13 

23.  Stakeman 13 

24.  Rear  chainman 14 

25.  Back  flag 14 

26.  Axeman 14 

27.  Leveler ;  also  form  of  notes 14 

V 


vi  Contents. 

SECTION  PAGE 

28.  Rodman 15 

29.  Topographer 16 

30.  Preliminary  by  stadia 17 

CHAPTER  III. 
LOCATION  SURVEY. 

31.  Nature  of  "location  " 18 

32.  First  method , .  18 

33.  Second  method 18 

34.  Long  tangents 19 

35.  Tangent  from  broken  line  of  preliminary 19 

36.  Method  of  staking  out  location 19 

CHAPTER  IV. 
SIMPLE  CURVES. 

37.  Definitions 20 

38.  Measurements 20 

39.  Degree  of  curve 20 

40-41.   Formulas  for  degree  and  radius 21 

42.  Approximate  method 21 

43.  Tangent  distance  T 22 

44.  Approximate  method 22 

45.  External  distance  E 23 

46.  Middle  ordinate  M 23 

47.  Chord  C , 23 

48.  Formulas  for  R  and  D  in  terms  of  T,  E,  M,  C,  1 24 

49.  Sub-chord  c 24 

50.  Sub-angle  d 24 

51.  Length  of  curve  L 26 

52.  Method  of  deflection  angles 27 

53.  Example ;  application  to  parabola 27 

54-55.  Deflection  angles  for  simple  curves 27-28 

56.  Field-work    29 

57.  Example 30 

58.  Caution 30 

59.  Field-work  when  curve  cannot  be  laid  out  from  P.C 31 

60.  Second  method 31 

61.  Field-work  of  finding  P.C.  and  P.T 32 

62.  Example 33 


Contents.  vii 


SECTION  PAGE 

63.  Form  of  transit  book  for  curves 34 

64.  Curves  with  metric  system , 35 

65.  Method  of  deflection  distances 36 

66.  Field-work   36 

67.  Approximate  computation  of  offsets 37 

68.  Offset  between  two  curves 37 

69.  Deflection  distances  for  curve  beginning  with  sub-chord. .  37 

70.  Example 38 

71.  Approximate  solution  of  right  triangles 38 

72.  Field-work  for  method  of  deflection  distances 39 

73.  Caution 39 

74.  Deflection  distances  when  first  sub-chord  is  short 40 

75.  Method  of  offsets  from  the  tangent 40 

76.  Field-work   40 

77-78.  Middle  ordinates 41 

79.  Ordinate  at  any  point 41 

80.  Middle  ordinate,  approximate  formula 42 

81.  Example 42 

82-83.  Any  ordinate,  approximate  formulas 42-43 

84.  Example 43 

85.  Find  a  series  of  points  by  middle  ordinates 44 

86-88.   Substitute  new  curves  to  end  in  parallel  tangents 11-46 

89-90.   Curve  to  join  tangents  and  pass  through  given  point. .  .46-47 
91-92.   Find  where  curve  and  given  line  intersect 47-48 

93.  Approximate  method 48 

94.  Find  tangent  from  curve  to  given  point 48 

95.  Approximate  method 49 

96-99.   Obstacles  in  cases  of  curves 50-51 


CHAPTER  V. 
COMPOUND  CURVES. 

100.  Definitions 52 

101.  Field-work    52 

102.  Data 53 

103.  Given  Ri,  Rs,  Ii,  L;  required/,    Ti,  Ts 53 

104.  Given  Ts,  Rs,  Ri,  /;    required  /„  Ii,  Ti 54 

105.  Given  Ts,  Rs,  L,  /;    required  Ii,  Ti,  Ri 54 

106.  Given  Ti,  Ts,  R,,  I;    required  Ii,  L,  Ri 54 

107.  Given  Ti,  Ri,  Rs,  I;    required"^,  It,    Ts 55 

108.  Given  Ti,  Ri,  Ii,  J;    required  L,  T,,  Rs 55 

109.  Given  Ti,  Ts,  Ri,  I;    required  L,  Ii,  Ra 55 


viii  Contents. 

SECTION  PAGE 

110.  Given,  long  chord,  angles,  and  R*;  required  Ii,  J«,  I,  Ei.  56 

111.  Given,  long  chord,  angles,  and  Ri;   required  Ii,  L,  'I,  Rs.  56 

112.  Substitute  for  simple,  a  compound  curve  to  end  in  par- 

allel tangent 56 

113.  Example 57 

114.  Change  P.C.C.  so  as  to  join  parallel  tangent 58 

115.  Substitute  for  simple,  a  symmetrical  curve  with  flattened 

ends , 59 

116.  Substitute  curve  with  flattened  ends  to  pass  through 

middle  point 60 

117.  Substitute  simple  curve  for  curves  with  connecting  tangent  61 

CHAPTER  VI. 
REVERSED  CURVES. 

118.  Use  of  reversed  curves 62 

119-122.  Between  parallel  tangents,  common  radius 62-63 

123-124.  Between  parallel  tangents,  unequal  radii 63-64 

125-126.  Find  common  radius  to  connect  tangents  not  parallel .  64-65 
127-128.  Find  Ilt  I2,  T2  when  /,  Tlt  RI,  R^  are  given 65-66 

CHAPTER  VII. 
PARABOLIC  CURVES. 

129.  Use  of  parabolic  curves 67 

130.  Properties  of  the  parabola 67 

131.  Lay  out  parabola  by  offsets  from  tangent 68 

132.  Field-work  69 

133.  Parabola  by  middle  ordinates 70 

134.  Vertical  curves,  where  used 70 

135.  Method  for  vertical  curve  200  feet  long 71 

136.  General  method 72 

137.  Example 74 

138.  To  find  proper  length  of  vertical  curve 74 

CHAPTER  VIII. 
TURNOUTS. 

139.  Definitions - 75 

140.  Find  frog  angle  from  number  of  frog 76 

141-142.  Descriptions  of  features  of  turnouts 77 


Contents.  ix 


SECTION  PAGE 

143.  Given,  gauge,  frog  angle,  and  throw;  required,  radius, 

length,  and  lead 78 

144.  Given,  gauge,   number,   and    throw;    required,   radius, 

length,  and  lead 79 

145.  Given,  gauge ;  required,  middle  ordinate 79 

146.  Find  angle  of  crotch  frog 80 

147.  Find  number  of  crotch  frog 80 

148.  Find  proper  radius  for  turnout  inside ;  also  lead 81 

149.  Approximate  formula  to  find  degree  of  turnout  inside  ...  82 
150-151.  Find  radius  of  turnout  outside ;  also  lead 83-84 

152.  Example  of  case,  §  148 84 

153.  Description  of  split  switch 85 

154.  Find  radius  of  turnout  for  split  switch 86 

155.  Complication  on  account  of  straight  frog 87 

156.  Find  radius  of  turnout  for  split  Switch  and  straight  frog.  87 

157.  Methods  of  connecting  parallel  tracks  by  turnouts 88 

158-162.  Formulas  for  these 88-89 

163.  Formulas  for  a  series  of  parallel  tracks 90 

164.  Find  radius  of  turnout  curve  from  frog  to  parallel  curved 

track  outside 90 

165.  Approximate  method 91 

166.  Example.    Precise  method 92 

167.  Approximate  method 92 

168.  Find  radius  of  turnout  curve  from  frog  to  parallel  curved 

track  inside 93 

169.  Special  case  for  turnout  outside 93 

170.  Calculations  for  turnout  between  parallel  curved  tracks.  94 

171.  Approximate  method 94 


CHAPTER  IX. 

"Y"  TRACKS  AND  CROSSINGS. 

172.  Definition 95 

173.  Main  track  tangent,  "Y"  track  curved,  and  turnout 

curved 95 

174.  Main  track  tangent,  "Y"  curved,  turnout  curved  with 

tangent 96 

175-176.  Main  track  tangent  and  curve  "  Y  "  curved,  turnout 

curved 96-97 

177.  Crossing  of  tangent  and  curve 97 

178.  Crossing  of  two  curves 98 


x  Contents. 

CHAPTER  X. 
CUBIC  SPIRAL  EASEMENT  CURVE... 

SECTION  PAGE 

179.  Necessity  for ;  also  elevation  of  outer  rail 99 

180.  Equations  for  cubic  parabola  and  cubic  .spiral 100-101 

181.  Discussion  of  character  of  easement  curves 101 

182.  Deflection  angles  and  spiral  angles 102 

183.  Values  of  y  in  terms  of  I  and  R .103 

184.  Values  of  p  and  q  in  terms  of  x,  y,  R,  and  s 103 

185-186.   Fieldwork  and  example  for  cubic  spiral 104 

187-188.  Deflection    angles    for    spiral    from    intermediate 

points 104-106 

189-190.   Laying  out  spiral  by  deflection  angles 106-107 

191.  Laying  out  spiral  by  offsets 108 

192.  Spirals  for  compound  cifves 108-109 

193.  Tangent  distances 109-110 

194-196.   Substitution  of  curve  and  spiral  for  simple  curve.  .110-112 

CHAPTER  XL 

SETTING  STAKES  FOR  EARTHWORK. 

197.  Data 113 

198.  What  stakes  and  how  marked 113 

199.  Method  of  finding  rod  reading  for  grade 114 

200.  Example 115 

201.  Cut  or  fill  at  center 115 

202.  Side  stake  for  level  section 116 

203-206.   Side  stakes  when  surface  is  not  level 116-118 

207.   Slope-board  or  level-board 118 

208-210.  Keeping  the  notes 119 

211-212.  Form  of  note-book 120-121 

213.   Cross-sections ;  where  taken 122 

214-215.   Passing  from  cut  to  fill 122-123 

216.  Opening  in  embankment 123 

217.  General  level  notes 123 

218-221.  Level,  three-level,  five-level,  irregular,  sections 124 

CHAPTER  XII. 
METHODS  OF  COMPUTING  EARTHWORK. 

222.  Principal  methods  used 125 

223.  Averaging  end  areas 125 


Contents.  xi 


SECTION  PAGE 

224.  Kinds  of  cross-sections  specified 126 

225.  Level  cross-section 126 

226.  Three-levef  section 127 

227.  Three-level  section ;  second  method 128 

228.  Five-level  section 129 

229.  Irregular  section 129 

230.  Planimeter 130 

231.  Comment  on  end  area  formula 130 

232.  Prismoidal  formula 130 

233.  Prismoidal  formula  for  prisms,  wedges,  pyramids 131 

234.  Nature  of  regular  section  of  earthwork 132 

235-237.   Prismoidal  formula  applied  where  upper  surface  is 

warped 132-134 

238.   Wide  application  of  prismoidal  formula 134 

239-240.    Prismoidal  correction 135-136 

241.  Where  applicable ;  also  special  case 137 

242.  Correction  for  pyramid 138 

243.  Correction  for  five-level  sections 138 

244-245.   Correction  for  irregular  sections 138-139 

246.  Value  of  prismoidal  correction 139 

247.  Method  of  middle  areas 140 

248.  Method  of  equivalent  level  sections 140 

249.  Method  of  mean  proportionals 140 

250.  Henck's  method 140 

251.  Formula 141 

252-254.   Example 142-143 

255-256.   Comment  on  Henck's  and  end  area  methods 143-144 

257-263.   Examples  comparing  the  various  methods 144-146 


CHAPTER  XIII. 
SPECIAL  PROBLEMS  IN  EARTHWORK. 

264.  Correction  for  curvature 147 

265.  Correction  where  chords  are  less  than  100  ft 149 

266.  Correction  of  irregular  sections 149 

267.  Opening  in  embankment 150 

268.  Borrow-pits.. . ; : 152 

269.  Truncated  triangular  prism 152 

270.  Truncated  rectangular  prism 153 

271.  Assembled  prisms 155 

272.  Additional  heights .  .156-157 


xii  Contents. 


CHAPTER  XIV. 

EARTHWORK  TABLES. 
SECTION  PAGE 

273.  Formula  for  use  in  tables 158 

274.  Arrangement  of  table 159 

275.  Explanation  of  table 159 

276-277.   Example  of  use,  including  prismoidal  correction  table  160 

278.  Prismoidal  correction  applied  for  section  less  than  100  feet  161 

279.  Tables,  where  published 161 

280.  Tables  of  triangular  prisms 161 

281.  Where  published 161 

282.  Arrangement  of  tables  of  triangular  prisms 162 

283.  Example  of  use 163 

284-285.  Application  to  irregular  sections 164 


CHAPTER  XV. 
EARTHWORK  DIAGRAMS. 

286-287.  Method  of  diagrams 165-166 

288.  Forms  of  equations  available  for  straight  lines 166 

289.  Method  of  use  of  diagrams 166 

290-291.  Computations  and  table  for  diagram  of  prismoidal 

correction 167-168 

292.  Diagram  for  prismoidal  correction  and  explanation  of 

construction 168-169 

293.  Explanation  and  example  of  use 170 

294.  Table  of  triangular  prisms 170 

295-298.   Computations  and  table  for  diagram  of  three-level 

sections '. 171-173 

299-300.  Checks  upon  computations 175 

301.  Explanation  of  diagram  ;  also  curve  of  level  section 175 

302.  Use  of  diagram  for  three-level  sections 176 

303.  Comment  on  rapidity  by  use  of  diagrams 177 

304.  Special  use  to  find  prismoidal  correction  for  irregular 

sections 177 


CHAPTER  XVI. 
HAUL. 

305.  Definition  and  measure  of  haul 178 

306-307.  Length  of  haul,  how  found. .  .*. 178-179 

308.   Formula  for  center  of  gravity  of  a  section 179 


Contents.  xiii 


SECTION  PAGE 

309-310.  Formula  deduced : 180-181 

311.  Formula  modified  for  use  with  tables  or  diagrams 182 

312.  For  section  less  than  100  feet 182 

313.  For  series  of  sections 183 


CHAPTER  XVII. 

MASS  DIAGRAM. 

314.  Definition 184 

315.  Table  and  method  of  computation 184-185 

316.  Mass  diagram  and  its  properties 186-187 

317.  Graphical  measure  of  haul  explained 187 

318.  Application  to  mass  diagram 188-189 

319.  Further  properties 189 

320.  Mass  diagram ;  showing  also  borrow  and  waste 190-191 

321.  Profitable  length  of  haul 191 

322-323.   Example  of  use  of  diagram 192-193 

TABLES  AND  DIAGRAMS...  ..194-206 


RAILROAD  CURVES  AND  EARTHWORK. 


CHAPTER  I. 

1.  The  operations  of  "locating"  a  railroad,  as  commonly 
practiced  in  this  country,  are  three  in  number  :  — 

I.   RECONNOISSANCE. 
II.  PRELIMINARY  SURVEY. 
III.    LOCATION  SURVEY. 

I.  RECONNOISSANCE. 

2.  The  Reconnoissance  is  a  rapid  survey,  or  rather  a  critical 
examination  of  country,  without  the  use  of  the  ordinary  instru- 
ments of  surveying.    Certain  instruments,  however,  are  used, 
the  Aneroid  Barometer,  for  instance.     It  is  very  commonly  the 
case  that  the  termini  of  the  railroad  are  fixed,  and  often  inter- 
mediate points  also.    It  is  desirable  that  no  unnecessary  re- 
strictions as  to  intermediate  points  should  be  imposed  on  the 
engineer  to  prevent  his  selecting  what  he  finds  to  be  the  best 
line,  and  for  this  reason  it  is  advisable  that  the  reconnoissance 
should,  where  possible,  precede  the  drawing  of  the  charter. 

3.  The  first  step  in  reconnoissance  should  be  to  procure  the 
best  available  maps  of  the  country  ;  a  study  of  these  will  gen- 
erally furnish  to  the  engineer  a  guide  as  to  the  routes  or  section 
of  country  that  should  be  examined.     If  maps  of  the  United 
States  Geological  Survey  are  at  hand,  with  contour  lines  and 
other  topography  carefully  shown,  the  reconnoissance  can  be 
largely  determined  upon  these  maps.    Lines  clearly  imprac- 
ticable will  be  thrown  out,  the  maximum  grade  closely  deter- 
mined, and  the  field  examinations  reduced  to  a  minimum*     No 

1 


2  Railroad  Curves  and  Earthwork. 

route  should  be  accepted  finally  from  any  such  map,  but  a 
careJfaL  field  examination  should  be  made  over  the  routes  indi- 
cated'ohi1  the,  tfortt'U1?;  maps.  The  examination,  in  general, 
should  cover  the  general  section  of  country,  rather  than  be 
confined  to  a  single  line  between  the  termini.  A  straight  line 
and  a  straight  grade  from  one  terminus  to  the  other  is  desirable, 
but  this  is  seldom  possible,  and  is  in  general  far  from  possible. 
If  a  single  line  only  is  examined,  and  this  is  found  to  be  nearly 
straight  throughout,  and  with  satisfactory  grades,  it  may  be 
thought  unnecessary  to  carry  the  examination  further.  It  will 
frequently,  however,  be  found  advantageous  to  deviate  con- 
siderably from  a  straight  line  in  order  to  secure  satisfactory 
grades.  In  many  cases  it  will  be  necessary  to  wind  about  more 
or  less  through  the  country  in  order  to  secure  the  best  line. 
Where  a  high  hill  or  a  mountain  lies  directly  between  the 
points,  it  may  be  expected  that  a  line  around  the  hill,  and 
somewhat  remote  from  a  direct  line,  will  prove  more  favorable 
than  any  other.  Unless  a  reasonably  direct  line  is  found,  the 
examination,  to  be  satisfactory,  should  embrace  all  the  section  of 
intervening  country,  and  all  feasible  lines  should  be  examined. 

4.  There  are  two  features  of  topography  that  are  likely  to 
prove  of  especial  interest  in  reconnoissance,  ridge  lines  and 
valley  lines. 

A  ridge  line  along  the  whole  of  its  course  is  higher  than  the 
ground  immediately  adjacent  to  it  on  each  side.  That  is,  the 
ground  slopes  downward  from  it  to  both  sides.  It  is  also  called 
a  watershed  line. 

A  valley  line,  to  the  contrary,  is  lower  than  the  ground  im- 
mediately adjacent  to  it  on  each  side.  The  ground  slopes 
upward  from  it  to  both  sides.  Valley  lines  may  be  called  water- 
course lines. 

A  pass  is  a  place  on  a  ridge  line  lower  than  any  neighboring 
points  on  the  same  ridge.  Very  important  points  to  be  deter- 
mined in  reconnoissance  are  the  passes  where  the  ridge  lines 
are  to  be  crossed ;  also  the  points  where  the  valleys  are  to  be 
crossed  ;  and  careful  attention  should  be  given  to  these  points. 
In  crossing  a  valley  through  which  a  large  stream  flows,  it  may 
be  of  great  importance  to  find  a  good  bridge  crossing.  In  some 
cases  where  there  are  serious  difficulties  in  crossing  a  ridge,  a 
tunnel  may  be  necessary.  Where  such  structures,  either 


Reconnaissance.  3 

bridges  or  tunnels,  are  to  be  built,  favorable  points  for  their 
construction  should  be  selected  and  the  rest  of  the  line  be  com- 
pelled to  conform.  In  many  parts  of  the  United  States  at  the 
present  time,  the  necessity  for  avoiding  grade  crossings  causes 
the  crossings  of  roads  and  streets  to  become  governing  points 
of  as  great  importance  as  ridges  and  valleys. 

5.  There  are  several  purposes  of  reconnoissance :   first,  to 
find  whether  there  is  any  satisfactory  line  between  the  proposed 
termini ;  if  so,  second,  to  establish  which  is  the  most  feasible  ; 
third,  to  determine  approximately  the  maximum  grade  neces- 
sary to  be  used ;    fourth,  to  report  upon  the   character   or 
geological  formation  of  the  country,  and  the  probable  cost  of 
construction  depending  somewhat  upon  that ;  fifth,  to  make 
note  of  the  existing  resources  of  the  country,  its  manufactures, 
mines,  agricultural  or  natural  products,  and  the  capabilities  for 
improvement  and  development  of  the  country  resulting  from 
the  introduction  of  the  railroad.    The  report  upon  reconnois- 
sance should  include  information  upon  all  these  points.     It  is 
for  the  determination  of  the  third  point  mentioned,  the  rate  of 
maximum  grade,  that  the  barometer  is  used.     Observing  the 
elevations  of  governing  points,  and  knowing  the  distances  be- 
tween those  points,  it  is  possible  to  form  a  good  judgment  as  to 
what  rate  of  maximum  grade  to  assume. 

6.  The  Elevations  are  usually  taken  by  the  Aneroid  Barome- 
ter.    Tables  for  converting  barometer  readings  into  elevations 
above  sea-level  are  readily  available  and  in  convenient  form  for 
field  use.     (See  Searles'  or  Henck's  Field  Books.) 

Distances  may  be  determined  with  sufficient  accuracy  in 
many  cases  from  the  map,  where  a  good  one  exists.  Where 
this  method  is  impossible  or  seems  undesirable,  the  distance 
may  be  determined  in  one  of  several  different  ways.  When 
the  trip  is  made  by  wagon,  it  is  customary  to  use  an  Odometer, 
an  instrument  which  measures  and  records  the  number  of 
revolutions  of  the  wheel  to  which  it  is  attached,  and  thus  the 
distance  traveled  by  the  wagon.  There  are  different  forms  of 
odometer.  In  its  most  common  form,  it  depends  upon  a  hang- 
ing weight  or  pendulum,  which  is  supposed  to  hold  its  position, 
hanging  vertical,  while  the  wheel  turns.  The  instrument  is 
attached  to  the  wheel  between  the  spokes  and  as  near  to  the 
hub  as  practicable.  At  low  speeds  it  registers  accurately  ;  as  the 


4  Railroad  Curves  and  Earthwork. 

speed  is  increased,  a  point  is  reached  where  the  centrifugal  force 
neutralizes  or  overcomes  the  force  of  gravity  upon  the  pendu- 
lum, and  the  instrument  fails  to  register  accurately,  or  perhaps 
at  high  speeds  to  register  at  all.  If  this  form  of  odometer  is 
used,  a  clear  understanding  should  be  had  of  the  conditions 
under  which  it  fails  to  correctly  register.  A  theoretical  discus- 
sion might  closely  establish  the  point  at  which  the  centrifugal 
force  will  balance  the  force  of  gravity.  The  wheel  striking 
against  stones  in  a  rough  road  will  create  disturbances  in  the 
action  of  the  pendulum,  so  that  the  odometer  will  fail  to  register 
accurately  at  speeds  less  than  that  determined  upon  the  above 
assumption. 

Another  form  of  odometer  is  manufactured  which  is  con- 
nected both  with  the  wheel  and  the  axle,  and  so  measures 
positively  the  relative  motion  between  the  wheel  and  axle, 
and  this  ought  to  be  reliable  for  registering  accurately.  Many 
engineers  prefer  to  count  the  revolutions  of  the  wheel  them- 
selves, tying  a  rag  to  the  wheel  to  make  a  conspicuous  mark 
for  counting. 

When  the  trip  is  made  on  foot,  pacing  will  give  satisfactory 
results.  An  instrument  called  the  Pedometer  registers  the 
results  of  pacing.  As  ordinarily  constructed,  the  graduations 
read  to  quarter  miles,  and  it  is  possible  to  estimate  to  one- 
tenth  that  distance.  Pedometers  are  also  made  which  register 
paces.  In  principle,  the  pedometer  depends  upon  the  fact  that, 
with  each  step,  a  certain  shock  or  jar  is  produced  as  the  heel 
strikes  the  ground,  and  each  shock  causes  the  instrument  to 
register.  Those  registering  miles  are  adjustable  to  the  length 
of  pace  of  the  wearer. 

If  the  trip  is  made  on  horseback,  it  is  found  possible  to  get 
good  results  with  a  steady-gaited  horse,  by  first  determining  his 
rate  of  travel  and  figuring  distance  by  the  time  consumed  in 
traveling.  Excellent  results  are  said  to  have  been  secured  in 
this  way. 

7.  It  is  customary  for  engineers  not  to  use  a  compass  in 
reconnoissance,  although  this  is  sometimes  done  in  order  to 
trace  the  line  traversed  upon  the  map,  and  with  greater  accu- 
racy. A  pocket  level  will  be  found  useful.  The  skillful  use  of 
pocket  instruments  will  almost  certainly  be  found  of  great  value 
to  the  engineer  of  reconnoissance. 


Reconnoissance.  5 

It  may,  in  cases,  occur  that  no  maps  of  any  value  are  in 
existence  or  procurable.  It  may  be  necessary,  in  such  a  case, 
to  make  a  rapid  instrumental  survey,  the  measurements  being 
taken  either  by  pacing,  chain,  or  stadia  measurements.  This 
is,  however,  unusual. 

8.  The  preliminary  survey  is  based  upon  the  results  of  the 
reconnoissance,  and  the  location  upon  the  results  of  the  pre- 
liminary survey.  The  reconnoissance  thus  forms  the  founda- 
tion upon  which  the  location  is  made.  Any  failure  to  find  a 
suitable  line  and  the  best  line  constitutes  a  defect  which  no 
amount  of  faithfulness  in  the  later  work  will  rectify.  The 
most  serious  errors  of  location  are  liable  to  be  due  to  imper- 
fect reconnoissance ;  an  inefficient  engineer  of  reconnoissance 
should  be  avoided  at  all  hazards.  In  the  case  of  a  new  railroad, 
it  would,  in  general,  be  proper  that  the  Chief  Engineer  should 
in  person  conduct  this  survey.  In  the  case  of  the  extension  of 
existing  lines,  this  might  be  impracticable  or  inadvisable,  but 
an  assistant  of  known  responsibility,  ability,  and  experience 
should  in  this  case  be  selected  to  attend  to  the  work. 


CHAPTEE   II. 
II.    PRELIMINARY  SURVEY. 

9.  The  Preliminary  Survey  is  based  upon  the  results  of  the 
reconnoissance.     It  is  a  survey  made  with  the  ordinary  instru- 
ments of  surveying.     Its  purpose  is  to  fix  and  mark  upon  the 
ground  a  first  trial  line  approximating  as  closely  to  the  proper 
final  line  as  the  difficulty  of  the  country  and  the  experience  of 
the  engineer  will  allow  ;  further  than  this,  to  collect  data  such 
that  this  survey  shall  serve  as  a  basis   upon  which  the   final 
Location  may  intelligently  be  made.     In  order  to  approximate 
closely  in  the  trial  line,  it  is  essential  that  the  maximum  grade 
should  be  determined  or  estimated  as  correctly  as  possible,  and 
the  line  fixed  with  due  regard  thereto. 

It  will  be  of  value  to  devote  some  attention  here  to  an  ex- 
planation about  Grades  and  "Maximum  Grades." 

10.  Grades.  —  The  ideal  line  in  railroad  location  is  a  straight 
and  level  line.    This  is  seldom,  if  ever,  realized.    When  the  two 
termini  are  at  different  elevations,  a  line  straight  and  of  uni- 
form grade  becomes  the  ideal.     It  is  commonly  impossible  to 
secure  a  line  of  uniform  grade  between  termini.     In  operating 
a  railroad,  an  engine  division  will  be  about  100  miles,  some- 
times less,  often  more.     In  locating  any  100  miles  of  railroad, 
it  is  almost  certain  that  a  uniform  grade  cannot  be  maintained. 
More  commonly  there  will  be  a  succession  of  hills,  part  of  the 
line  up  grade,  part  down  grade.     Sometimes  there  will  be  a 
continuous  up  grade,  but  not  at  a  uniform  rate.     With  a  uni- 
form grade,  a  locomotive  engine  will  be  constantly  exerting  its 
maximum  pull  or  doing  its  maximum  work  in  hauling  the 
longest  train  it  is  capable  of  hauling ;  there  will  be  no  power 
wasted  in  hauling  a  light  train  over  low  or  level  grades  upon 
which  a  heavier  train  could  be  hauled.     Where  the  grades  are 
not  uniform,  but  are  rising  or  falling,  or  rising  irregularly,  it 
will  be  found  that  the  topography  on  some  particular  o  or  10 

6 


Preliminary  Survey.  7 

miles  is  of  such  a  character  that  the  grade  here  must  be  steeper 
than  is  really  necessary  anywhere  else  on  the  line  ;  or  there 
may  be  two  or  three  stretches  of  grade  where  about  the  same 
rate  of  grade  is  necessary,  steeper  than  elsewhere  required. 
The  steep  grade  thus  found  necessary  at  some  special  point  or 
points  on  the  line  of  railroad  is  called  the  "  Maximum  Grade" 
or  "Ruling  Grade"  or  "Limiting  Grade,"  it' being  the  grade 
that  limits  the  weight  of  train  that  an  engine  can  haul  over  the 
whole  division.  It  should  then  be  the  effort  to  make  the  rate 
of  maximum  grade  as  low  as  possible,  because  the  lower  the 
rate  of  the  maximum  grade,  the  heavier  the  train  a  given  loco- 
motive can  haul,  and  because  it  costs  not  very  much  more  to 
haul  a  heavy  train  than  a  light  one.  The  maximum  grade 
determined  by  the  reconnoissance  should  be  used  as  the  basis 
for  the  preliminary  survey.  How  will  this  affect  the  line? 
Whenever  a  hill  is  encountered,  if  the  maximum  grade  be 
steep,  it  may  be  possible  to  carry  the  line  straight,  and  over 
the  hill ;  if  the  maximum  grade  be  low,  it  may  be  necessary  to 
deflect  the  line  and  carry  it  around  the  hill.  When  the  maxi- 
mum grade  has  been  once  properly  determined,  if  any  saving 
can  be  accomplished  by  using  it  rather  than  a  grade  less  steep, 
the  maximum  grade  should  be  used.  It  is  possible  that  the 
train  loads  will  not  be  uniform  throughout  the  division.  It 
will  be  advantageous  to  spend  a  small  sum  of  money  to  keep 
any  grade  lower  than  the  maximum,  in  view  of  the  possibility 
that  at  this  particular  point  the  train  load  will  be  heavier  than 
elsewhere  on  the  division.  Any  saving  made  will  in  general 
be  of  one  or  more  of  three  kinds  :  — 

a.    Amount  or  "  quantity  "  of  excavation  or  embankment ; 

6.    Distance  ; 

c.    Curvature. 

11.  In  some  cases,  a  satisfactory  grade,  a  low  grade  for  a 
maximum,   can  be  maintained  throughout  a  division   of  100 
miles  in  length,  with  the  exception  of  2  or  3  miles  at  one  point 
only.     So  great  is  the  value  of  a  low  maximum  grade  that  all 
kinds  of  expedients  will  be  sought  for,  to  pass  the  difficulty 
without  increasing  the  rate  of  maximum  grade,  which  we  know 
will  apply  to  the  whole  division. 

12.  Sometimes  by  increasing  the  length  of  line,  we  are  able 
to  reach  a  given  elevation  with  a  lower  rate  of  grade.    Some- 


8  Railroad  Curves  and  Earthwork. 

times  heavy  and  expensive  cuts  and  fills  may  serve  the  pur- 
pose. Sometimes  all  such  devices  fail,  and -there  still  remains 
necessary  an  increase  of  grade  at  this  one  point,  but  at  this 
point  only.  In  such  case  it  is  now  customary  to  adopt  the 
higher  rate  of  grade  for  these  2  or  3  miles  and  operate  them  by 
using  an  extra  or  additional  engine.  In  this  case,  the  "ruling 
grade  "  for  the  division  of  100  miles  is  properly  the  "maximum 
grade"  prevailing  over  the  division  generally,  the  higher  grade 
for  a  few  miles  only  being  known  as  an  "  Auxiliary  Grade  "  or 
more  commonly  a  "  Pusher  Grade."  The  train  which  is  hauled 
over  the  engine  division  is  helped  over  the  auxiliary  or  pusher 
grade  by  the  use  of  an  additional  engine  called  a  "Pusher." 
Where  the  use  of  a  short  "  Pusher  Grade  "  will  allow  the  use 
of  a  low  "  maximum  grade,"  there  is  evident  economy  in  its 
use.  The  critical  discussion  of  the  importance  or  value  of 
saving  distance,  curvature,  rise  and  fall,  and  maximum  grade, 
is  not  within  the  scope  of  this  book,  and  the  reader  is  referred 
to  Wellington's  "Economic  Theory  of  Railway  Location." 

13.  The  Preliminary  Survey  follows  the  general  line  marked 
out  by  the  reconnoissance,  but  this  rapid  examination  of  coun- 
try may  not  have  fully  determined  which  of  two  or  more  lines 
is  the  best,  the  advantages  may  be  so  nearly  balanced.    In  this 
case  two  or  more  preliminary  surveys  must  be  made  for  com- 
parison.    When  the  reconnoissance  has  fully  determined  the 
general  route,  certain  details  are  still  left  for  the  preliminary 
survey  to  determine.     It  may  be  necessary  to  run  two  lines, 
one  on  each  side  of  a  small  stream,  and  possibly  a  line  crossing 
it  several  times.    The  reconnoissance  would  often  fail  to  settle 
minor  points  like  this.      It  is  desirable  that  the  preliminary 
survey  should  closely  approximate  to  the  final  line,  but  it  is 
not  important  that  it  should  fully  coincide  anywhere. 

An  important  purpose  of  the  "preliminary"  is  to  provide  a 
map  which  shall  show  enough  of  the  topography  of  the  country, 
so  that  the  Location  proper  may  be  projected  upon  this  map. 
Working  from  the  line  of  survey  as  a  base  line,  measurements 
should  be  taken  sufficient  to  show  streams  and  various  natural 
objects  as  well  as  the  contours  of  the  surface. 

14.  The  Preliminary  Survey  serves  several  purposes  :  — 
First.   To  fix  accurately  the    maximum    grade  for  use  in 

Location. 


Preliminary  Survey.  9 

Second.   To  determine  which  of  several  lines  is  best. 
Third.  To  provide  a  map  as  a  basis  upon  which  the  Location 
can  properly  be  made. 

Fourth.   To  make  a  close  estimate  of  the  cost  of  the  work. 
Fifth.   To  secure,  in  certain  cases,  legal  rights  by  filing  plans. 

15.  It  should  be  understood  that  the  preliminary  survey 
is,  in  general,  simply  a  means  to  an  end,  and  rapidity  and 
economy  are  desirable.     It  is  an  instrumental  survey.     Meas- 
urements of  distance  are  taken  usually  with  the  chain,  although 
a  tape  is  sometimes  used.    Angles  are  taken  generally  with  a 
transit ;    some  advocate  the  use  of  a  compass.     The  line  is 
ordinarily  run  as  a  broken  line  with  angles,  but  is  occasionally 
run  with  curves  connecting  the  straight  stretches,  generally  for 
the  reason  that  a  map  of  such  a  line  is  available  for  filing,  and 
certain  legal  rights  result  from  such  a  filing.     With  a  compass, 
no  backsight  need  be  taken,  and,  in  passing  small  obstacles,  a 
compass  will  save  time  on  this  account.    A  transit  line  can  be 
carried  past  an  obstacle  readily  by  a  zigzag  line.     Common 
practice  among  engineers  favors  the  use  of  the  transit  rather 
than  the  compass.    Stakes  are  set,  at  every  "  Station,"  100  feet 
apart,  and  the  stakes  are  marked  on  the  face,  the  first  0,  the 
next  I ,  then  2,  and  so  to  the  end  of  the  line.    A  stake  set  1025 
feet  from  the  beginning  would  be  marked  10  +  25. 

Levels  are  taken  on  the  ground  at  the  side  of  the  stakes,  and 
as  much  oftener  as  there  is  any  change  in  the  inclination  of  the 
ground.  All  the  surface  heights  are  platted  on  a  profile,  and 
the  grade  line  adjusted. 

16.  The  line  should  be  run  from  a  governing  point  towards 
country  allowing  a  choice  of  location,  that  is  from  a  pass  or 
from  an  important  bridge  crossing,  towards  country  offering  no 
great  difficulties.     There  is  an  advantage  in  running  from  a 
summit  downhill,  subject,  however,  to  the   above  considera- 
tions.    In  running  from  a  summit  down  at  a  prescribed  rate 
of  grade,  an  experienced  engineer  will  carry  the  line  so  that,  at 
the  end  of  a  day's  work,  the  levels  will  show  the  line  to  be 
about  where  it  ought  to  be.     For  this  purpose,  the  levels  must 
be  worked  up  and  the  profile   platted  to  date  at  the  close  of 
each  day.     Any  slight  change  of  line  found  necessary  can  then 
be  made  early  the  next  morning.    A  method  sometimes  adopted 
in  working  down  from  a  summit  is  for  the  locating  engineer  to 


10  Railroad  Curves  and  Earthwork. 

plat  his  grade  line  on  the  profile,  daily  in  advance,  and  then 
during  the  day,  plat  a  point  on  his  profile  whenever  he  caii 
conveniently  get  one  from  his  leveler,  and  thus  find  whether 
his  line  is  too  high  or  too  low. 

17.  Occasionally  the  result  of  two  or  three  days'  work  will 
yield  a  line  extremely  unsatisfactory,  enough  so  that  the  work 
of  these  two  or  three  days  will  be  abandoned.  The  party 
"backs  up"  and  takes  a  fresh  start  from  some  convenient 
point.  In  such  case  the  custom  is  not  to  tear  out  several 
pages  of  note-book,  but  instead  to  simply  draw  a  line  across 
the  page  and  mark  the  page  "Abandoned."  At  some  future 
time  the  abandoned  notes  may  convey  useful  information  to 
the  effect  that  this  line  was  attempted  and  found  unavailable. 
In  general,  all  notes  worth  taking  are  worth  saving. 

Sometimes  after  a  line  has  been  run  through  a  section  of 
country,  there  is  later  found  a  shorter  or  better  line. 


g    3 

, 


In  the  figure  used  for  illustration,  the  first  line,  "  A  "  Line, 
is  represented  by  AEBCD,  upon  which  the  stations  are  marked 
continuously  from  A  to  D,  850  stations.  The  new  line,  "  B  " 
Line,  starts  from  E,  Sta.  102  +  60,  and  the  stationing  is  held 
continuous  from  0  to  where  it  connects  with  the  "A"  Line  at 
C.  The  point  C  is  Sta.  312  +  27  of  the  "A"  Line,  and  is  also 
Sta.  307  -f  13  of  the  "  B"  Line.  It  is  not  customary  to  restake 
the  line  from  C  to  D  in  accordance  with  "B"  Line  stationing. 
Instead  of  this,  a  note  is  made  in  the  note-books  as  follows  :  — 
Sta.  312  +  27  "  A  "  Line  =  307  +  13  "  B  "  Line. 

Some  engineers  make  the  note  in  the  following  form  :  — 
Sta.  307  to  3 13  =  86  ft. 

The  first  form  is  preferable,  being  more  direct  and  less  liable 
to  cause  confusion. 


Preliminary  Survey.  11 

18.  All  notes  should  be  kept  clearly  and  nicely  in  a  note- 
book—  never  on  small  pieces  of  paper.      The  date  and  the 
names  of  members  of  the  party  should  be  entered  each  day  in 
the  upper  left-hand  corner  of  the  page.     An  office  copy  should 
be  made  as  soon  as  opportunity  offers,  both  for  safety  and  con- 
venience.    The  original  notes  should  always  be  preserved;  they 
would  be  admissible  as  evidence  in  a  court  of  law  where  a  copy 
would  be  rejected.     When  two  or  more  separate  or  alternate 
lines  are  run,  they  may  be  designated 

Line  "  A,"  Line  "  B,"  Line  "  C," 
or  "A"  Line,  "B"  Line,  "C"  Line. 

19.  The  Organization  of  Party  may  be  as  follows :  — 

1.  Locating  Engineer. 

2.  Transitman. 

3.  Head  Chainman. 

4.  Stakeman.  Transit  Party. 

5.  Rear  Chainman. 

6.  Back  Flag. 

7.  Axemen  (one  or  more) . 

8.  Leveler.  )  T       ,  ,. 

.     \  Level  Party. 

9.  Rodman  (sometimes  two).  ) 

10.  Topographer.  {  Topographical  Party 

11.  Assistant. 

12.  Cook. 

13.  Teamster.     . 

20.  The  Locating  Engineer  is  the  chief  of  party,  and  is 
responsible   for  the   business  management  of  the  camp   and 
party,  as  well  as  for  the 'conduct  of  the  survey.     He  deter- 
mines where  the  line  shall  run,  keeping  ahead  of  the  transit, 
and  establishing  points  as  foresights  or  turning-points  for  the 
transitman.     In  open  country,  the  extra  axeman  can  assist  by 
holding  the  flag  at  turning-points,  and  thus  allowing  the  locat- 
ing engineer  to  push  on  and  pick  out  other  points  in  advance. 
The  locating  engineer  keeps  a  special  note-book  or  memorandum 
book  ;  in  it  he  notes  on  the  ground  the  quality  of  material,  rock, 
earth,  or  whatever  it  may  be  ;   takes  notes  to  determine  the 
lengths  and  positions  of  bridges,  culverts,  and  other  structures  ; 
shows  the  localities  of  timber,  building  stones,  borrow  pits,  and 


12 


Railroad  Curves  and  Earthwork. 


other  materials  valuable  for  the  execution  of  the  work  ;  in  fact, 
makes  notes  of  all  matters  not  properly  attended  to  by  the 
transit,  leveling,  or  topography  party.  The  rapid  and  faithful 
prosecution  of  the  work  depend  upon  the  locating  engineer, 
and  the  party  ought  to  derive  inspiration  from  the  energy  and 
vigor  of  their  chief,  who  should  be  the  leader  in  the  work.  In 
open  and  easy  country,  the  locating  engineer  may  instill  life 
into  the  party  by  himself  taking  the  place  of  the  head  chain- 
man  occasionally.  In  country  of  some  difficulty,  his  time  will 
be  far  better  employed  in  prospecting  for  the  best  line. 

21.  The  Transitman  does  the  transit  work,  ranges  in  the 
line  from  the  instrument,  measures  the  angles,  and  keeps  the 
notes  of  the  transit  survey.  The  following  is  a  good  form  for 
the  left-hand  page  of  the  note-book  :  — 


Station 

Point 

Deflection 

Observed 
Bearing 

Calculated 
Bearing 

7 

N    3°30'E 

N    3°38'E 

6 

0  +  24 

33°02'R 

5 

N29°30'W 

N29°24'W 

4 

0 

I2°09'L 

3 

2 

1 

N  I7°I5'W 

N  I7°I5'W 

0 

0 

Notes  of  topography  and  remarks  are  entered  on  the  right- 
hand  page,  which,  for  convenience,  is  divided  into  small  squares 
by  blue  lines,  with  a  red  line  running  up  and  down  through  the 
middle. 

The  stations  run  from  bottom  to  top  of  page.  The  bearing  is 
taken  at  each  setting  and  recorded  just  above  the  corresponding 
point  in  the  note-book,  or  opposite  a  part  of  the  line,  rather 
than  opposite  the  point.  Ordinarily,  the  transitman  takes  the 
bearings  of  all  fences  and  roads  crossed  by  the  line,  finds  the 
stations  from  the  rear  chainman,  and  records  them  in  their 
proper  place  and  direction  on  the  right-hand  page  of  the  note- 
book. Section  lines  of  the  United  States  Land  Surveys  should  be 


Preliminary  Survey.  13 

observed  in  the  same  way.  The  transitman  is  next  in  authority 
to  the  locating  engineer,  and  directs  the  work  when  the  latter 
is  not  immediately  present.  The  transitman,  while  moving 
from  point  to  point,  setting  up,  and  ranging  line,  limits  the 
speed  of  the  entire  party,  and  should  waste  no  time. 

22.  The  Head  Chainman  carries  a  "flag"  and  the  forward 
end  of  the  chain,  which  should  be  held  level  and  firm  with  one 
hand,  while  the  flag  is  moved  into  line  with  the  other.     He 
should  always  put  himself  nearly  in  line  before  receiving  a 
signal  from  the  transitman ;   plumbing  may  be  done  with  the 
flag.   When  the  point  is  found,  the  stakeman  will  set  the  stake. 
When  a  suitable  place  for  a  turning-point  is  reached,  a  signal 
should  be  given  the  transitman  to  that  effect.     A  nail  should  be 
set  in  top  of  the  stake  at  all  turning-points.    A  proper  under- 
standing should  be  had  with  the  transitman  as  to  signals. 

Signals  from  the  Transitman. 

A  horizontal  movement  of  the  hand  indicates  that  the  rod 
should  be  moved  as  directed. 

A  swinging  movement  of  the  hand,  "Plumb  the  rod  as 
indicated." 

A  movement  of  both  hands,  or  waving  the  handkerchief 
freely  above  the  head,  means  "All  right." 

At  long  distances,  a  handkerchief  can  be  seen  to  advantage  ; 
when  snow  is  on  the  ground,  something  black  is  better. 

Signals  from  the  Head  Chainman. 

Setting  the  bottom  of  flag  on  the  ground  and  waving  the  top, 
means  "  Give  the  line." 

Raising  the  flag  above  the  head  and  holding  it  horizontal 
with  both  hands  :  "  Give  line  for  a  turning-point." 

The  "all  right"  signal  is  the  same  as  from  the  transitman. 

In  all  measurements  less  than  100  feet  (or  a  full  chain),  the 
head  chainman  holds  the  end  of  the  chain,  leaving  the  reading 
of  the  measurement  to  the  rear  chainman. 

The  head  chainman  regulates  the  speed  of  the  party  during 
the  time  that  the  instrument  is  in  place,  and  should  keep  alive 
all  the  time.  The  rear  chainman  will  keep  up  as  a  matter  of 
necessity. 

23.  The  Stakeman  carries,  marks,  and  drives  the  stakes  at 
the  points  indicated  by  the  head  chainman.     The  stakes  should 


14  Railroad  Curves  and  Earthwork. 

be  driven  with  the  flat  side  towards  the  instrument,  and  marked 
on  the  front  with  the  number  of  the  station.  Intermediate 
stakes  should  be  marked  with  the  number  of  the  last  station 
+  the  additional  distance  in  feet  and  tenths,  as  10  +  67.4.  The 
stationing  is  not  interrupted  and  taken  up  anew  at  each  turning- 
point,  but  is  continuous  from  beginning  to  end  of  the  survey. 
At  each  turning-point  a  plug  should  be  driven  nearly  flush  with 
the  ground,  and  a  witness  stake  driven,  in  an  inclined  position, 
at  a  distance  of  about  15  inches  from  the  plug,  and  at  the  side 
towards  which  the  advance  line  deflects,  and  marked  W  and 
under  it  the  station  of  the  plug. 

24.  The  Rear  Chainman  holds  the  rear  end  of  the  chain 
over  the  stake  last  set,  but  does  not  hold  against  the  stake  to 
loosen  it.     He  calls  "  Chain"  each  time  when  the  new  stake  is 
reached,  being  careful  not  to  overstep  the  distance.     He  should 
stand  beside  the  line  (not  on  it)  when  measuring,  and  take  pains 
not  to  obstruct  the  view  of  the  transitman.     He  checks,  and  is 
responsible  for  the  correct  numbering  of  stakes,  and  for  all 
distances  less  than  100  feet,  as  the  head  chainman  always  holds 
the  end  of  the  chain.    The  stations  where  the  line  crosses  fences, 
roads,  and  streams  should  be  set  down  in  a  small  note- book,  and 
reported  to  the  transitman  at  the  earliest  convenient  opportu- 
nity.    The  rear  chainman  is  responsible  for  the  chain. 

25.  The  Back  Flag  holds  the  flag  as  a  backsight  at  the 
point  last  occupied  by  the  transit.     The  only  signals  necessary 
for  him  to  understand  from  the  transitman  are   "plumb  the 
flag"  and  "all  right."     The  flag  should  always  be  in  position, 
and  the  transitman  should  not  be  delayed  an  instant.     The 
back  flag  should  be  ready  to  come  up  the  instant  he  receives 
the  "all  right"  signal  from  the  transitman.     The  duties  are 
simple,  but  frequently  are  not  well  performed. 

26.  The  Axeman  cuts  and  clears  through  forest  or  brush. 
A  good  axeman  should  be  able  to  keep  the  line  well,  so  as  to 
cut  nothing  unnecessary.     In  open  country,  he  prepares  the 
stakes  ready  for  the  stakeman  or  assists  the  locating  engineer 
as  fore  flag. 

27.  The  Leveler  handles  the  level  and  generally  keeps  the 
notes,  which  may  have  the  following  form  for  the  left-hand 
page.     The  right-hand  page  is  for  remarks  and  descriptions  of 
turning-points  and  bench-marks.     It  is  desirable  that  turning- 


Preliminary  Survey. 


15 


Station 

+  s 

H  1 

-  s 

Elevation 

B.M. 

4.67 

104.67 

100.00 

0 

5.7 

99.0 

1 

6.9 

97.8 

2 

3.4 

101.3 

T.P. 

9.26 

112.81 

1.12 

103.55 

3 

8.5 

104.3 

points  should,  where  possible,  be  described,  and  that  all  bench- 
marks should  be  used  as  turning-points.  Readings  on  turning- 
points  should  be  recorded  to  hundredths  or  to  thousandths  of 
a  foot,  dependent  upon  the  judgment  of  the  Chief  Engineer. 
Surface  readings  should  be  made  to  the  nearest  tenth,  and  ele- 
vations set  down  to  nearest  tenth  only.  A  self-reading  rod  has 
advantages  over  a  target  rod  for  short  sights.  A  target  rod  is 
possibly  better  for  long  sights  and  for  turning-points.  The 
"Philadelphia  Rod"  is  both  a  target  rod  and  a  self-reading 
rod,  and  is  thus  well  adapted  for  railroad  use.  Bench-marks 
should  be  taken  at  distances  of  from  1000  to  1500  feet,  depending 
upon  the  country.  All  bench-marks,  as  soon  as  calculated, 
should  be  entered  together  on  a  special  page  near  the  end  of 
the  book.  The  leveler  should  test  his  level  frequently  to  see 
that  it  is  in  adjustment.  The  leveler  and  rodman  should 
together  bring  the  notes  to  date  every  evening  and  plat  the 
profile  to  correspond. 

The  profile  of  the  preliminary  line  should  show  :  — 

a.  Surface  line  (in  black) . 

b.  Grade  line  (in  red). 

c.  Grade  elevations  at  each  change  in  grade  (in  red). 

d.  Rate  of  grade,  per  100  (in  red);  rise  +,  fall  — . 

e.  Station  and  deflection  at  each  angle  in  the  line  (in  black). 
/.   Notes  of  roads,  ditches,  streams,  bridges,  etc.  (in  black) . 

28.  The  Rodman  carries  the  rod  and  holds  it  vertical  upon 
the  ground  at  each  station  and  at  such  intermediate  points  as 
mark  any  important  change  of  slope  of  the  ground.  The  sur- 
face of  streams  and  ponds  should  be  taken  when  met,  and  at 
frequent  intervals  where  possible,  if  they  continue  near  the  line, 


16  Railroad  Curves  and  Earthwork. 

Levels  should  also  be  taken  of  high-water  marks  wherever 
traces  of  these  are  visible.  The  rodman  carries  a  small  note- 
book in  which  he  enters  the  rod  readings  at  all  turning-points. 
In  country  which  is  open,  but  not  level,  the  transit  party  is 
liable  to  outrun  the  level  party.  In  such  cases  greater  speed 
will  be  secured  by  the  use  of  two  rodmen. 

29.  The  Topographer  is,  or  should  be,  one  of  the  most  val- 
uable members  of  the  party.  In  times  past  it  has  not  always 
been  found  necessary  to  have  a  topographer,  or  if  employed, 
his  duty  has  been  to  sketch  in  the  general  features  necessary  to 
make  an  attractive  map,  and  represent  hills  and  buildings  suffi- 
ciently well  with  reference  to  the  line  to  show,  in  a  general 
way,  the  reason  for  the  location  adopted.  Sometimes  the  chief 
of  the  party  has  for  this  purpose  taken  the  topography.  At 
present  the  best  practice  favors  the  taking  of  accurate  data  by 
the  topography  party. 

The  topographer  (with  one  or  two  assistants)  should  take  the 
station  and  bearing  (or  angle)  of  every  fence  or  street  line 
crossed  by  the  survey  (unless  taken  by  the  transit  party) ;  also 
take  measurements  and  bearings  for  platting  all  fences  and 
buildings  near  enough  to  influence  the  position  of  the  Location; 
also  sketch,  as  well  as  may  be,  fences,  buildings,  and  other 
topographical  features  of  interest  which  are  too  remote  to  re- 
quire exact  measurement ;  and  finally,  establish  the  position  of 
contour  lines,  streams,  and  ponds,  within  limits  such  that  the 
Location  may  be  well  and  fully  determined  in  the  contoured 
map. 

The  work  of  taking  contours  is  accomplished  by  the  use  of 
hand  level  and  tape  (pacing  may,  in  many  cases,  be  sufficient). 
The  elevation  at  the  center  line  at  each  station  is  found  from 
the  leveler.  Contours  are  generally  taken  at  vertical  intervals 
of  5  feet.  The  contour  line,  say  to  the  right  of  the  station,  is 
found  by  reading  upon  a  light  hand-marked  rod  the  difference 
in  elevation  between  the  center  line  at  the  stake,  and  the  re- 
quired contour.  A  cloth  tape,  held  by  the  assistant  topog- 
rapher, will  serve  the  purpose  of  a  rod.  The  next  contour  to 
the  right  can  be  readily  found,  owing  to  the  fact  that  the  topog- 
rapher's eye  is  nearly  5  feet  above  the  ground,  making  leveling 
easy.  An  allowance  should  be  made  for  the  difference  between 
5  feet  and  the  height  of  the  eye.  Sections  both  right  and  left 


Preliminary  Survey.  17 

should  be  taken  as  often  as  necessary,  the  distances  to  each  con- 
tour measured,  and  the  lines  between  the  points  thus  determined 
sketched  in  on  the  ground.  Books  of  convenient  size  are  made 
and  divided  into  small  cross-sections  to  facilitate  sketching. 
Cross-section  blocks  or  pads  will  be  considered  equally  good  by 
some  engineers.  The  distance  to  which  contour  lines  should  be 
taken  depends  on  the  character  of  the  country.  The  object 
should  be  to  take  contours  as  far  from  the  line  as  is  necessary 
in  order  to  furnish  contours  requisite  for  determining  the  posi- 
tion of  the  located  line. 

Instead  of  a  hand  level,  some  engineers  use  a  clinometer  and 
take  and  record  side  slopes. 

Topography  can  be  taken  rapidly  and  well  by  stadia  survey 
or  by  plane  table.  This  is  seldom  done,  as  most  engineers  are 
not  sufficiently  familiar  with  their  use. 

30.  Some  engineers  advocate  making  a  general  topographi- 
cal survey  of  the  route  by  stadia,  instead  of  the  survey  above 
described.  In  this  case  no  staking  out  by  "  stations  "  would  be 
done.  All  points  occupied  by  the  transit  should  be  marked  by 
plugs,  which  can  be  used  to  aid  in  marking  the  Location  on 
the  ground  after  it  is  determined  on  the  contour  map.  This 
method  has  been  used  a  number  of  times,  and  is  claimed  to  give 
economical  and  satisfactory  results ;  it  is  probable  that  it  will 
have  constantly  increasing  use  in  the  future,  and  will  prove 
the  best  method  in  a  large  share  of  cases. 


CHAPTER   III. 
III.    LOCATION  SURVEY. 

31.  The  Location  Survey  is  the  final  fitting  of  the  line  to 
the  ground.    In  Location,  curves  are  used  to  connect  the  straight 
lines  or  "tangents,"  and  the  alignment  is  laid  out  complete, 
ready  for  construction. 

The  party  is  much  the  same  as  in  the  preliminary,  and  the 
duties  substantially  the  same.  More  work  devolves  upon  the 
transitman  on  account  of  the  curves,  and  more  skill  is  useful 
in  the  head  chainman  in  putting  himself  in  position  on  curves. 
He  can  readily  range  himself  on  tangent.  The  form  of  notes 
will  be  shown  later.  The  profile  is  the  same,  except  that  it 
shows,  for  alignment  notes,  the  P.C.  and  P.T.  of  curves,  and  also 
the  degree  and  central  angle,  and  whether  to  the  right  or  left. 

It  is  well  to  connect  frequently  location  stakes  with  prelimi- 
nary stakes,  when  convenient,  as  a  check  on  the  work. 

In  making  the  location  survey,  two  distinct  methods  are  in 
use  among  engineers  :  — 

32.  First  Method  of  Location.  — Use  preliminary  survey  and 
preliminary  profile  as  guides  in  reading  the  country,  and  locate 
the  line  upon  the  ground.    Experience  in  such  work  will  enable 
an  engineer  to  get  very  satisfactory  results  in  this  way,  in 
nearly  all  cases.     The  best  engineers,  in  locating  in  this  way, 
as  a  rule  lay  the  tangents  first,  and  connect  the  curves  after- 
wards.    It  will  appear  later  how  this  is  done. 

33.  Second  Method.  —  Use  preliminary  line,  preliminary  pro- 
file, and  especially  the  contour  lines  on  the  preliminary  map ; 
make  a  paper  location,  and  run  this  in  on  the  ground.     Some 
go  so  far  as  to  give  their  locating  engineer  a  complete  set  of 
notes  to  run  by.     This  is  going  too  far.     Whether  it  is  best  to 
go  farther  than  to  fix,  on  the  map,  the  location  of  tangents, 
and  specify  the  degree  of  curve,  is  a  question.    A  conservative 
method  is  to  do  no  more  than  this,  and  in  some  cases,  leave 

18 


Location  Survey.  19 

the  degree  of  curve  even  an  open  question.  The  second  method 
is  gaining  in  favor,  but  the  first  method  is,  even  now,  much 
used.  It  is  well  accepted,  among  engineers,  that  no  reversed 
curve  should  be  used ;  200  feet  of  tangent,  at  least,  should  inter- 
vene. Neither  should  any  curve  be  very  short,  say  less  than 
300  feet  in  length. 

34.  A  most  difficult  matter  is  the  laying  of  a  long  tangent, 
so  that  it  shall  be  straight.     Lack  of  perfect  adjustment  and 
construction  of  instrument  will  cause  a  "swing"  in  the  tan- 
gent.    The  best  way  is  to  run  for  a  distant  foresight.     Another 
way  is  to  have  the  transit  as  well  adjusted  as  possible,  and  even 
then  change  ends  every  time  in  reversing,  so  that  errors  shall 
not  accumulate.     It  will  be  noticed  that  the  preliminary  is  run 
in  without  curves  because  more  economical  in  time  ;  sometimes 
curves  are  run  however,  either  because  the  line  can  be  run 
closer  to  its  proper  position,  or  sometimes  in  order  to  allow  of 
filing  plans  with  the  United  States  or  separate  States. 

35.  In  Location,  a  single  tangent  often  takes  the  place  of  a 
broken  line   in  the  preliminary,  and  it  becomes  important  to 
determine  the  direction  of  the  tangent  with  reference  to  some 
part  of  the  broken  line.     This  is  readily  done  by  finding  the 
coordinates  of  any  given  point  with  reference  to  that  part  of 
the  broken  line    assumed    temporarily   as  a  meridian.     The 
course  of  each  line  is  calculated,  and  the  coordinates  of  any 
point  thus  found.     It  simplifies  the  calculation  to  use  some 
part  of  the  preliminary  as  an  assumed  meridian,  rather  than  to 
use  the  actual  bearings  of  the  lines.     The  coordinates  of  two 
points  on    the  proposed   tangent  allow  the    direction  of   the 
tangent  to  be  determined  writh  reference  to  any  part  of  the 
preliminary.     "When  the  angles  are  small,  an  approximation 
sufficiently  close  will  be  secured,  by  assuming  in  all  cases  that 
the  cosine  of  the  angle  is  1.000000  and  that  the  sines  are  directly 
proportional  to  the  angles  themselves.     In  addition  to  this,  take 
the  distances  at  the   nearest  even  foot,    and  the   calculation 
becomes  much  simplified. 

36.  The  located  line,  or  "Location,"  as  it  is  often  called,  is 
staked  out  ordinarily  by  center  stakes  which  mark  a  succession 
of  straight  lines,  connected  by  curves  to  which  the  straight  lines 
are  tangent.     The  straight  lines  are  by  general  usage  called 
"Tangents." 


CHAPTER  IV. 
SIMPLE  CURVES. 

37.  The  curves  most  generally  in  use  are  circular  curves,  al- 
though parabolic  and  other  curves  are  sometimes  used.    Circular 
curves  may  be  classed  as  Simple,  Compound,  Reversed,  or  Spiral. 

A  Simple  Curve  is  a  circular  arc,  extending  from  one  tan- 
gent to  the  next.  The  point  where  the  curve  leaves  the  first 
tangent  is  called  the  "P.O.,"  meaning  the  point  of  curvature, 
and  the  point  where  the  curve  joins  the  second  tangent  is 
called  the  "  P.  T.,"  meaning  the  point  of  tangency.  The  P.  C. 
and  P.T.  are  often  called  the  Tangent  Points.  If  the  tan- 
gents be  produced,  they  will  meet  in  a  point  of  intersection 
called  the  "Vertex,"  V.  The  distance  from  the  vertex  to  the 
P.O.  or  P.T.  is  called  the  "Tangent  Distance,"  T.  The  dis- 
tance from  the  vertex  to  the  curve  (measured  towards  the 
center)  is  called  the  External  Distance,  E.  The  line  joining 
the  middle  of,  the  Chord,  (7,  with  the  middle  of  the  curve  sub- 
tended by  this  chord,  is  called  the  Middle  Ordinate,  M.  The 
radius  of  the  curve  is  called  the  Radius,  B.  The  angle  of 
deflection  between  the  tangents  is  called  the  Intersection  Angle, 
/.  The  angle  at  the  center  subtended  by  a  chord  of  100  feet  is 
called  the  Degree  of  Curve,  D.  A  chord  of  less  than  100  feet 
is  called  a  sub-chord,  c ;  its  central  angle  a  sub-angle,  d. 

38.  The  measurements  on  a  curve  are  made  : 

(a)  from  P.  C.  by  a  sub-chord  (sometimes  a  full  chord  of 
100  ft.)  to  the  next  even  station,  then 

(6)  by  chords  of  100  feet  each  between  even  stations,  and 
finally, 

(c)  from  the  last  station  on  the  curve,  by  a  sub-chord  (some- 
times a  full  chord  of  100  ft.)  to  P.T.  The  total  distance  from 
P.  C.  to  P.  T.  measured  in  this  way,  is  the  Length  of  Curve,  L. 

39.  The  degree  of  curve  is  here  defined  as  the  angle  sub- 
tended by  a  chord  of  100  feet,  rather  than  by  an  arc  of  100  feet. 

20 


Simple   Curves. 


21 


Either  assumption  involves  the  use  of  approximate  methods 
either  in  calculations  or  measurements,  if  the  convenient  and 
customary  methods  are  followed.  It  is  believed  that  on  the 
merits  of  the  question,  it  is  best  to  accept  the  definition  given, 
and  the  practice  in  this  country  is  largely  in  harmony  with  this 
definition. 

Outside  of  the  United  States  a  curve  is  generally  designated 
by  its  Radius,  E.  In  the  United  States  for  railroad  purposes, 
a  curve  is  generally  designated  by  its  Degree,  D. 

40.   Problem.     Given  E. 

Required  D. 


Bin    D  = 


(1) 


41.   Problem.     Given  D. 

Required  E. 

Example.     Given  D  =  1°. 

E  =     50  50  lo& 


50 


1.698970 


sin  \D     0°  30'  log  sin  7.940842 
RI  =  5729.6  log        3.758128 

42.   Problem.     Given  RI  (radius  of  1°  curve)  or  D\. 

Required  Ea  (radius  of  any  given  curve  of 

degree  =  Z>a). 
r,  50  50 


Eg  _  sin 


In  the  case  of  small  angles,  the  angles  are  proportional  to  the 
sines  (approximately), 


But  RI  =  5730  to  nearest  foot, 


Example. 


(4) 


Rlo  =  573.7  by  (3),  or  by  Allen's  Table  I. 
=  573.0  by  (4)  (approx.) 


22 


Railroad  Curves  and  Earthwork. 


Some  engineers  use  shorter  chords  for  sharp  curves,  as  1°  to 
7°,  100  ft. ;  8°  to  15°,  50  ft. ;  16°  to  20°,  25  ft.  Ea  =  ^^  is 
then  very  closely  approximate.  a 

Values  of  E  and  D  are  readily  convertible.  Table  V., 
Allen,  serves  this  purpose,  giving  accurate  results  or  values. 
In  problems  later,  where  either  E  or  D  is  given,  both  will,  in 
general,  be  assumed  to  be  given.  Approximate  values  can  be 
found  without  tables  by  (4).  The  radius  of  a  1°  curve  =  5730 
should  be  remembered. 

43.   Problem.     Given  7,  also  R  or  D. 
Required  T. 

AOB  =  NVB  =  7 
A  V         N 


A0  =  OB  =  72 
AV  =  VB   -T 


I  (5) 

Example.    Given  D  =  9;  I  =  60°  48'. 
Required  TQ. 

Table!.,       729  log  =  2.804327 
30°  24'  log  tan  =  9.  768414 

Tg  =  373.9  log  2.  572741 


44.   Approximate  Method. 

TI  =  RI  tan  %  7;  Ta  =  Ra  tan 


=        (approx.) 


(6) 


Table  III.,  Allen,  gives  values  of  T\  for  various  values  of  7. 
Table  IV.,  Allen,  gives  a  correction  to  be  added  after  divid- 
ing by  Da. 


Simple  Curves.  23 

Example.    As  before.     Given  D  =  9  ;  /  =  60°  48'. 
Required  T9. 

Table  III.,  Z\  60°  48'  =  3361.6(9 

Tg  =    373.51  (approx.) 

Table  IV.,  correction,  9°  and  61°  =         .38 

T<>  -   373.9    (exact) 

the  same  as  before 

45.  Problem.     Given  /,  also  R  or  D. 

Required  E. 
Using  previous  figure, 

VH  =  E  =  It  exsec  J  /  (7) 

Table  XIX.  gives  natural  exsec. 
Table  XV.  gives  logarithmic  exsec. 
Approximate  Method. 
By  method  used  for  (6), 

^«  =  f*  (approx.)  (8) 

Table  V.  gives  values  for  E\. 

46.  Problem.     Given  7,  also  R  or  D. 

Required  M. 

FH  =  M=RveTS$I  (9) 

Table  XIX.  gives  natural  vers. 
Table  XV.  gives  logarithmic  vers. 
Table  II.  gives  certain  middle  ordinates. 

47.  Problem.     Given  /,  also  R  or  D. 

Required  chord  AB  =  C. 

C  =  2  R  sin  %  I  (10) 

Table  VIII.  gives  values  for  certain  long  chords. 


24  Railroad  Curves  and  Earthwork. 

48.    Transposing,  we  find  additional  formulas,  as  follows  : 

from  (5)      R=  Tcot^I  (11) 

^  *=^ki  (12) 

W    *  =  dfi/  (13) 

(10)      S  =  -^—  (14) 


(4)     £a  =  ^°  (appro*.)  (15) 

-tia 

'(6)     Da=^(approx.)  (16) 

(8)     Da  =  Q-  (approx.)  (17) 

49.  Problem.     Given  sub-angle  d,  also  R  or  D. 
Required  sub-chord  c. 


(18) 
Approximate  Method. 


_c_      sin|^      _^  (approx.  )  (19) 

100  ^ 


The  precise  formula  is  seldom  if  ever  used. 

50.   Problem.     Given  sub-chord  c,  also  R  or  D. 
Required  sub-angle  d. 


2      100   2 


(20) 

(21) 


Simple  Curves.  25 


A  modification  of  this  formula  is  as  follows  : 

d_  cD 
2     200 

for  D  =  1 


for  any  value  Da 

-  =  c  x  0.3'  x  Da  (value  in  minutes)          (22) 

This  gives  a  very  simple  and  rapid  method  of  finding  the 
value  of  -  in  minutes,  and  the  formula  should  be  remembered, 

Example.     Given  sub-chord  =  63.7.    D  -  6°  30'. 
Required  sub-  angle  d  (or  -  ]• 

I.   By  (20)     63.7  II.   By  (21)  63.7 


3.25  =~ 


3185 
3822 
414.05 

4°.  14 

d  =  4°  08' 


III.   By  (22)     63.7 
0.3 


19.11 
6.5 
9555 
11466 
124.215  minutes 


=  2°  04' 


Method  III.  seems  preferable  to  I.  or  II. 


26  Railroad  Curves  and  Earthwork. 

51.    Problem.     Given  I  and  D. 
Required  L. 

(a)  When  the  P.  C.  is  at  an  even  station,  D  will  be  contained 
in  /a  certain  number  of  times  n,  and  there  will  remain  a  sub- 
angle  d  subtended  by  its  chord  c. 


100—  =  =  100  n  +    c  =  L  (approx.) 

(6)  When  the  P.  (7.  is  at  a  sub-station  the  same  reasoning 
holds,  and 

L  -  100  —  (approx.)  (23) 

Transposing, 

J  =        (approx.)  (24) 


(25) 

These  formulas  (23)  (24)  (25),  though  approximate,  are  the 
formulas  in  common  use. 

Example.     Given  7°  curve.     7=39°  37'. 
Required  L. 

1=      39°  37 
9  D  =  7)39.6107° 

5.6595  + 
L  =       566.0 

Example.     Given-D  and  L. 
Required  I. 
Given  8°  curve 

also,  P.  T.  =  93+70.1 

P.O.  =  86  +  49.3 

L=    1      20.8 

D  =  _  8 

57.664 

60' 


39.84 
1=     57°  40' 


Simple  Curves. 


27 


N 


52.   Method  of  Deflection  Angles. 

If  at  any  point  on  an  existing  curve  a  tangent  to  the  curve  be 
taken,  the  angles  from  the  tangent  to  any  given  points  on  the 
curve  may  be  measured,  and  the  angles  thus  found  may  be 
called  Total  Deflections  to  those  points  (as  NAl,  NA2,  NA3). 

In  laying  out  successive  points  upon  a  straight  line  (as  on  a 
"Tangent "),  each  point  is  generally  fixed  by 
(a)  measurement  from  the 

preceding  point  and         A 

(6)  line;  * 

the  line  on  a  tangent  will 
be  the  same  for  all  points. 

Similarly,  in  laying  out 
a  curve,  successive  points 
may  be  fixed  by 
(a)  measurement  from  the 
preceding  point  and 
(6)  line  ; 

the  line  in  this  case,  for  the.  curve,  will  be  that  found  by  using 
the  total  deflection  calculated  for  each  point.  In  the  figure  pre- 
ceding, the  chord  distance  A I  and  the  total  deflection  NAl  fix 
point  I  ;  the  chord  distance  1-2  and  total  deflection  NA2  fix 
point  2  ;  and  2-3  and  NA3  fix  3.  A  curve  can  be  conveniently 
laid  out  by  this  method  if  the  proper  total  deflections  can  be 
readily  computed. 


~    100'  -<    100'  * 


53.  Problem.     Given  a  Parabolic  Curve. 

Required  total  deflections  to 
B-C-D  and  chord  lengths 
AB-BC-CD. 

Give  results  to  the  nearest 
minute  or  nearest  ^  foot. 

54.  Simple  Curves. 

In  the  case  of  "  Simple  Curves,"  the  "total  deflections"  can 
be  readily  computed,  and  the  method  of  "  deflection  angles  "  is 
therefore  well  adapted  to  laying  them  out. 


28  Railroad  Curves  and  Earthwork. 

55.   Problem.     To  find  the  Total  Deflections  for  a  Simple 
Curve  having  given  the  Degree. 

I.    When  the  curve  begins  and  ends  at  even  stations. 

The  distance  from  station  to  station  is  100  feet.    The  deflec- 
tion angles  are  required. 


An  acute  angle  between  a  tangent  and  a  chord  is  equal  to 
one  half  the  central  angle  subtended  by  that  chord 

A  I  =  100  VAl=iD 

The  acute  angle  between  two  chords  having  their  vertices  in 
the  circumference  is  equal  to  one  half  the  arc  included  between 
those  chords. 

I  -  2  =  100  and    I  A  2  =  £  D     Similarly, 

2  -  3  =  100  and   2  A  3  =  \  D 

3  -  B  =  100  and   3  A  B  =  £  D 

This  angle  |  D  is  called  by  Henck  and  Searles  the  Deflection 
Angle,  and  will  be  so  called  here.     Shunk  and  Trautwine  call 
it  the  u  Tangential  Angle."    The  weight  of  engineering  opinion 
appears  to  be  largely  in  favojrpf  the  "  Deflection  Angle." 
The  "  Total  Deflections"  will  be  as  follows: 
VA  I  =\D 
VA2  =  VA  I  +    D 


VAB  will  be  found  by  successive  increments  of  \  D. 
VAB  =  VBA  =  £7.     This  furnishes  a  "  check  "  on  the  compu- 
tation. 
II.    When  the  curve  begins  and  ends  with  a  sub-chord. 

VA  I  =\d 

VA2  =  VA  I  +D 


Simple  Curves.  29 

VAB  is  found  by  adding  |  da  to  previous  "  total  deflection." 
VAB  =  VB A  =  £  J.     This  furnishes  « '  check. ' '     The  total  deflec- 
tions should  be  calculated  by  successive  increments ;  the  final 
"check"   upon  \I  then  checks  all    the  intermediate    total 
deflections.     The  example  on  next  page  will  illustrate  this. 

56.   Field-work  of  laying  out  a  simple  curve  having  given  the 
position  and  station  of  P.  C.  and  P.  T. 
(a)    Set  the  transit  at  P.  C.  (A). 
(6)    Set  the  vernier  at  0. 

(c)  Set  cross  hairs  on  V  (or  on  N  and  reverse). 

(d)  Set  off  %di (sometimes  f  7))  for  point  I. 

(e)  Measure  distance  ci  (sometimes  100)  and  fix  I. 
(/)  Set  off  total  deflection  for  point  2. 

($r)  Measure  distance  1-2  =  100  and  fix  2,  etc. 

(h)  When  total  deflection  to  B  is  figured,  see  that  it  =  $  7, 
thus  "checking"  calculations. 

(i)  See  that  the  proper  calculated  distance  c%  and  the  total 
deflection  to  B  agree  with  the  actual  measurements  on  the 
ground,  checking  the  field-work. 

(&)  Move  transit  to  P.  T.  (B). 

(Z)  Turn  vernier  back  to  0,  and  beyond  0  to  £7. 

(m)  Sight  on  A. 

(n)  Turn  vernier  to  0. 

(o)  Sight  towards  V  (or  reverse  and  sight  towards  P) ,  and  see 
that  the  line  checks  on  V  or  P. 

It  should  be  observed  that  three  "checks"  on  the  work  are 
obtained.  . 

1.  The  calculation  of  the  total  deflections  is  checked  if  total 
deflection  to  B  =  £  7. 

2.  The  chaining  is  checked  if  the  final  sub-chord  measured 
on  the  ground  =  calculated  distance. 

3.  The  transit  work  is  checked  if  the  total  deflection  to  B 
brings  the  line  accurately  on  B. 

The  check  in  I  is  effective  only  when  the  total  deflection  for 
each  point  is  found  by  adding  the  proper  angle  to  that  for  the 
preceding  point. 

The  check  in  3  assures  the  general  accuracy  of  the  transit 
work,  but  does  not  prevent  an  error  in  laying  off  the  total 
deflection  at  an  intermediate  point  on  the  curve. 


30  Railroad  Curves  and  Earthwork. 

57.   Example.     Given  Notes  of  Curve 

P.T.  13  +  45.0 
P.  C.  10  +  74.0 

Required  the  "  total  deflections  " 
to  sta.  11  ci  =     26 


7.8 

,7          -^° 

21  =  46.8  =  0°  47'  to  11 
2  3° 


C2=     45  3°  47'  to  12 

.3  3° 

13.5  6°  47'  to  13 

° 


8°  08'  to  13  +  45 
13  +  45.0 
10  +  74.0 
2      71.0  =  L 

6° 
16.26  16° 

60' 
15.6'  16' 

16°  16'  =  I 
8°  08'  =  i/  "check" 

58.   Caution. 

If  a  curve  of  nearly  180°  =  /  is  to  be  laid  out  from  A,  it  is 
evident  that  it  would  be  difficult  or  impossible  to  set  the  last 
point  accurately,  as  the  "intersection"  would  be  bad.  It  is 
undesirable  to  use  a  total  deflection  greater  than  30°. 

It  may  be  impossible  to  see  the  entire  curve  from  the  P.  C. 
at  A. 

It  will,  therefore,  frequently  happen  that  from  one  cause  or 
another  the  entire  curve  cannot  be  laid  out  from  {he  P.C.,  and 
it  will  be  necessary  to  use  a  modification  of  the  method  de- 
scribed above. 


Simple  Curves.  31 

59.  Field-work.  When  the  entire  curve  cannot  be  laid  out 
from  the  P.O. 

First  Method. 

(a)  Lay  out  curve  as  far  as  C,  as  before. 

(6)  Set  transit  point  at  some  convenient  point,  as  C  (even 
station  preferably) . 

(c)  Move  transit  to  C. 

(<Z)  Turn  vernier  back  to  0,  and  beyond  0  to  measure  the 
angle  VAC. 

(e)  Sight  on  A. 

(/)  Turn  vernier  to  0.  See  that  transit  line  is  on  auxiliary 
tangent  NCM  (VAC  =  NCA  being  measured  by  \  arc  AC). 

(gr)  Set  off  new  deflection  angle  (\d  or  £Z>). 

(A)  Set  point  4,  and  proceed  as  in  ordinary  cases. 


\ 
\ 
\ 

i 


60.    Second  Method. 

(a)  Set  point  C  as  before,  and  move  transit  to  C. 

(6)  Set  vernier  at  0. 

(c)  Sight  on  A. 

(d)  Set  off  the  proper  "  total  deflection  "  for  the  point  4= VA  4. 
NCA  +  MC4  =  VA4,  each  measured  by  \  arc  AC  4. 

(e)  Reverse  transit  and  set  point  4. 

(/)  Set  off  and  use  the  proper  "total  deflections"  for  the 
remaining  points. 

The  second  method  is  in  some  respects  more  simple,  as  the 
notes  and  calculations,  and  also  setting  off  angles,  are  the  same 
as  if  no  additional  setting  were  made.  By  the  first  method  the 
deflection  angles  to  be  laid  off  will,  in  general,  be  even  minutes, 


32  Railroad  Curves  and  Earthwork. 

often  degrees  or  half  degrees,  and  are  thus  easier  to  lay  off.  It 
is  a  matter  of  personal  choice  which  of  the  two  methods  shall 
be  used.  It  will  be  disastrous  to  attempt  an  incorrect  combina- 
tion of  parts  of  the  two  methods. 

61.   Field-work  of  finding  P.O.  and  P.T. 

In  running  in  the  line,  it  is  common  and  considered  advisable 
to  establish  "F,"  determine  the  station  of  "F,"  and  measure 
the  angle  J.  Having  given  /  only,  an  infinite  number  of  curves 
could  be  used.  It  is,  therefore,  necessary  to  assume  additional 
data  to  determine  what  curve  to  use.  It  is  common  to  proceed 
as  follows : 

(a)  Assume  either  (1)  JO  directly. 

(2)  E  and  calculate  D. 

(3)  T  and  calculate  D. 

It  is  often  difficult  to  determine  off-hand  what  degree  of  curve 
will  well  fit  the  ground.  Frequently  the  value  of  Ea  can  be 
readily  determined  on  the  ground.  The  determination  of  D 
from  Ea  is  readily  made,  using  the  approximate  formula 

Da  =  — •     Similarly,  we  may  be  limited  to  a  given  (or  ascer- 

Ea  T 

tain  able)  value  of  Ta,  and  from  this  readily  find  Da=^> 

J.  a 

The  value  of  Da  adopted  will,  in  general,  be  taken  to  the 
nearest  £°  (perhaps  only  to  nearest  degree)  rather  than  at  the 
exact  value  found,  as  above.  (Some  engineers  use  1°  40'  =  100' 
and  3°  20'  =  200',  etc.,  rather  than  1°  30'  or  3°  30',  etc.). 

(&)  From  the  data  finally  adopted  T  is  calculated  anew. 

(c)  The  instrument  still  being  at  F,  the  P.  T.  is  set  by  laying 
off  T. 

(d}  The  station  of  P.  C.  is  calculated  and  P.  C.  set. 

(e)  The  length  of  curve  L  is  calculated,  and  station  of  P.  T. 
thus  determined  (not  by  adding  T  to  station  of  F). 

Total  deflections  should  be  all  calculated  and  entered  in  note- 
book. 

Whether  j9,  E,  or  T  shall  be  assumed  depends  upon  the 
special  requirements  in  each  case.  Curves  are  often  run  out 
from  P.  C.  without  finding  or  using  F,  but  the  best  engineering 
usage  seems  to  be  in  favor  of  setting  F,  whenever  this  is  at  all 
practicable,  and  from  this  finding  the  P.C.  and  P.T. 


Simple  Curves. 


33 


62.   Example.     Given  a  Zme,  as  shown  in  sketch. 

Required  a  Simple  Curve  to    connect  the 
Tangents. 


P.  T.  is  to  be  at  least  400  ft.  from  end  of  line. 

Use  smallest  degree  or  half  degree  consistent  with  this. 

Find  degree  of  curve  and  stations  of  P.  C.  and  P.  T. 


Table  II., 


V-  46  +  72.7 
T  4  +  60.7 

P.  C.  42  +  12.0 
L  9  +  09.3 

P.T.      51  +  21.3 


22°  44'  930 

400 

1151.8      /  530  =  Ta 
1060         ^2.2  - 
918 

use  2°  30'  curve 
Ti  =  1151.8       /2.6° 
100  U60.7 

151 
150 
180 

Table  IV.,  correction         0 

460.7  =  T. 


22°  44'  =  / 
2.5°)22°  7333 

909.3  -L 


34  Railroad  Curves  and  Earthwork. 

63.   Form  of  Transit  Book  (left-hand  page). 


(Date) 

(Names  of  Party) 

Station 

Points 

Descrip.  of 
Curve 

Total 
Deflect. 

Observed 
Course 

114 

113 

112 

III 

110 

109 

O+90.0P.T. 

11°  15' 

N46°00'  E 

108 

E  =  1146.3 

9°  00' 

107 
106 

O+  68.0  V 

L=    450.0 
T=    228.0 

6°  30' 
4°  00' 

105 

O+40.0P.C. 

7=22°  30' 
5°  Right 

1°30' 

104 

103 

102 

101 

100 

99 

N  23°  15'  E 

98 

V  is  not  a  point  on  the  curve.  Nevertheless,  it  is  customary 
to  record  the  station  found  by  chaining  along  the  tangent. 

The  right-hand  page  is  used  for  survey  notes  of  crossings  of 
fences  and  various  similar  data.  It  seems  unnecessary  to  show 
a  sample  here. 


Simple  Curves.  35 

64.   Metric  Curves. 

In  Railroad  Location  under  the  "Metric  System"  a  chain  of 
100  meters  is  too  long,  and  a  chain  of  10  meters  is  too  short. 
Some  engineers  have  used  the  30-meter  chain,  some  the  25- 
meter  chain,  but  lately  the  20-meter  chain  has  been  generally 
adopted  as  the  most  satisfactory.  Under  this  system  a  "  Sta- 
tion" is  10  meters.  Ordinarily,  every  second  station  only  is 
set,  and  these  are  marked  Sta.  0,  Sta.  2,  Sta.  4,  etc.  On  curves, 
chords  of  20  meters  are  used.  Usage  among  engineers  varies  as 
to  what  is  meant  by  the  Degree  of  Curve  under  the  metric 
system.  There  are  two  distinct  systems  used,  as  shown  below. 

I.  The  Degree  of  Curve  is  the  angle  at  the  center  subtended 
by  a  chord  of  1  chain  of  20  meters. 

II.  The  Degree  of  Curve  is  the  deflection  angle  for  a  chord 
of  1  chain  of  20  meters  (or  one  half  the  angle  at  the  center). 

II.  Or,  very  closely,  the  Degree  of  Curve  is  the  angle  at  the 
center  subtended  by  a  chord  of  10  meters  (equal  to  1  station 
length) . 

For  several  reasons  the  latter  system  is  favored  here.  Tables 
upon  this  basis  have  been  calculated,  giving  certain  data  for 
metric  curves.  Such  tables  are  to  be  found  in  Allen's  Field 
and  Office  Tables. 

In  many  countries  where  the  metric  system  is  used,  it  is  not 
customary  to  use  the  Degree  of  Curve,  as  indicated  here.  In 
Mexico,  where  the  metric  system  is  adopted  as  the  only  legal 
standard,  very  many  of  the  railroads  have  been  built  by  com- 
panies incorporated  in  this  country,  and  under  the  direction  of 
engineers  trained  here.  The  usage  indicated  above  has  been 
the  result  of  these  conditions.  If  the  metric  system  shall  in 
the  future  become  the  only  legal  system  in  the  United  States, 
as  now  seems  possible,  one  of  the  systems  outlined  above  will 
probably  prevail. 

In  foreign  countries  where  the  Degree  of  Curve  is  not  used,  it 
is  customary,  as  previously  stated,  to  designate  the  curve  by 
its  radius  U,  and  to  use  even  figures,  as  a  radius  of  1000  feet, 
or  2000  feet,  or  1000  meters,  or  2000  meters.  As  the  radius  is 
seldom  measured  on  the  ground,  the  only  convenience  in  even 
figures  is  in  platting,  while  there  is  a  constantly  recurring  incon- 
venience in  laying  off  the  angles. 


36 


Railroad  Curves  and  Earthwork. 


65.    Problem.     Given  D  and  the  stations  of  P.  C.  and  P.  T. 
Required  to  lay  out  the  curve  by  the  method 
of  Deflection  Distances. 

I.    When  the  curve  begins  and  ends  at  even  stations. 

In  the  curve  AB,  let  * 

__A  E          N 

AN  be  a  tangent 
AE      any  chord  =  c 
EE'  perp.  to  AE'  =  a  = 
"tangent  deflection" 
FF  =  BB'  =  the 
"chord  deflection" 
AO  =  EO  =  R 

Draw    OM    perpen- 
dicular to  AE. 

Then 


(26) 


a  = 

FF'  =  2  a  ;  AF'  =  AE  produced 
When  AE  is  a  full  station  of  100  feet, 
1002 


66.   Field-work. 

The  P.  C.  and  P.  T.  are  assumed  to  have  been  set. 
(a)    Calculate  aioo- 

(6)    Set  point  E  distant  100  ft.  from  A  and  distant  aioo  from 
AE'  (AE'  <  100  ft.  ;  AE'E  =  90°). 

(c)  Produce  AE  to   F'  (EF  =  100  ft.),  and  find  F  distant 
2  aioo  from  F'  (EF  =  100  ft). 

(d)  Proceed  similarly  until  B  is  reached  (P.T.). 


Simple  Curves. 


37 


(e)    At   station    preceding    B    (P.T.)    lay    off    FG'  =  aioo 
(FG'B  =  90°). 
(/)    G'B  is  tangent  to  the  curve  at  B  (P.T.). 

67.  The  tangent  deflection  for  100  ft.  for  a  1°  curve  is  nearly 
0.875,  or  |  ft.    For  any  number  of  even  stations  n,  the  offset 
will  be 

an  =  |  n2  for  a  1°  curve  (approx.). 

an  =  |  n?Da  for  any  curve  (approx.)  (27) 

68.  Problem.     Given  two  Curves  of  degree  Z),  and  Df. 

Required  the  offset  between  the  two  Curves 
for  any  number  of  even  stations. 

From  (27)  am-  = 


(28) 


an  =  %  n2  (Df  -  Di)  (approx.) 


69.   Problem.     Given  D  and  the  stations  of  P.  C.  and  P.  T. 
Required  to  lay  out  the  Curve  by  Deflection 
Distances. 

II.   When  the  curve  begins  and  ends  with  a  sub-chord. 


\ 

by  (26) 


Let  AE  =  initial  sub-chord  =  c,- 
HB  =  final  sub-chord  =  c/ 
E'E  =  tang.  defl.  for  c,-  =  at 
H"H  =     "        "      "   c,  =  a. 


«:„»  = 


1002 
— 


at :  «ioo  =  ci2 : 1002  a<  =  a\ 

df :  OIQO  =  c/2  : 1002  o/  =  a\ 

In  general  it  is  better  to  use  (29)  than  a*  =  -^— 


(29) 


38  Railroad  Curves  and  Earthwork. 

70.  Example.     Given  P.  T.    20  +  42    Q0  R 

P.  C.     16  +  25 

Required  all  data  necessary  to  lay  out  curve  by  "  Deflec- 
tion Distances." 

Calculate  without  Tables.     Result  to  T^  foot. 
Radius  1°  curve  =  5730(6 
6°  955 

10Q2  1910)10000(5.235+ 

~  2  x  955  955 

=  5.24  ~450 

OQO 

O  f*  —_   1 A   A*7  Go,« 

«75  =  0.752  x  5.24  680 

=  2.95  5ZL 

«42  =  0.422  x  5.24 

=  0.92 
Table  II.,  aioo  =  5.234  (precise  vamp). 

71.  The  distance  AE'  is  slightly  shorter  than  AE.     It  is  gen- 
erally sufficient  to  take  the  point  E'  by  inspection  simply.     If 
desired  for  this  or  any  other  purpose,  a  simple  approximate 
solution  of  right  triangles  is  as  follows : 

Problem.     Given  the  hypotheneuse  (or  base}  and  altitude. 

Required  the  difference  between  base  and  hypothe- 
neuse, or  in  the  figure,  c  —  a. 

C2  _  a2  =  ft2 

(c  —  a)  (c  +  <?)  =  ft2 


c  -a 


c+a  (30) 

c  -a  =  -*-{approx.) 

2a 

Wherever  h  is  small  in  comparison  with  a  or  c,  the  approxi- 
mation is  good  for  ordinary  purposes. 

Example.  c  =  100     ft  =  10 

c-a  =  m=   O-50 
a        =  99.50 

The  precise  formula  gives  99.499. 


Simple  Curves.  39 

72.  Field-work  for  Case  II.  ,  p.  37. 

(art  Calculate  aioo,  «*,  «/•  Remember  that  tangent  deflec- 
tions are  as  the  squares  of  the  chords. 

aioo  may  be  found  generally  in  Table  II.,  Allen,  as  "tan- 
gent offset." 

(6)  Find  the  point  E,  distant  a,  from  AE'  and  distant  ct  from 
A.  (AE'E  =  90°.) 

(c)  Erect  auxiliary  tangent  at  E  (lay  off  AA'  =  af). 

(«")  From  auxiliary  tangent  A'E  produced,  find  point  F. 
(FF'  =  aioo;    EF  =  100  ;   EF'F  =  90°). 

(e)  From  chord  EF  produced,  find  point  G. 

(GG'  =  2  aioo  ;   FG'  =  FG  =  100). 

(/)  Similarly,  for  each  full  station,  use  2  aioo,  etc. 

(gr)  At  last  even  station  on  curve,  H,  erect  an  auxiliary  tan- 
gent (lay  off  GG"  =  «ioo  ;  GG"H  =  90°). 

(ft)  From  G"H  produced,  find  B  (B'B  =  a/,  etc.). 

(0  Find  tangent  at  B  (HH"  =  a,;  HH"B  =  90°). 

The  values  of  aioo,  a*,  a/,  should  be  calculated  to  the  nearest 
Tfcr  foot. 

73.  Caution.    The  tangent  deflections  vary  as  the  squares  of 
the  chords,  not  directly  as  the  chords. 

Curves  may  be  laid  out  by  this  method  without  a  transit  by 
the  use  of  plumb  line  or  "  flag"  for  sighting  in  points,  and  with 
fair  degree  of  accuracy. 

For  calculating  aioo,  a,-,  a/,  it  is  sufficient  in  most  cases  to  use 


the  approx.  value  Ea  =  —'—.     A  curve  may  be  thus  laid  out 

Da 
without  the  use  of  transit  or  tables. 

For  many  approximate  purposes  it  is  well  and  useful  to 
remember  that  the  "chord  deflection"  for  1°  curve  is  1.75  ft. 
nearly,  and  for  other  degrees  in  direct  proportion.  A  head 
chainman  may  thus  put  himself  nearly  in  line  without  the  aid 
of  the  transitman. 

The  method  of  "Deflection  Distances"  is  not  well  adapted 
for  common  use,  but  will  often  be  of  value  in  emergencies. 


40 


Railroad  Curves  and  Earthwork. 


74.    Problem.     Given  D  and  stations  of  P.C.  and  P.  T. 

Required  to  lay  out  the  curve  by  "  Deflection 
Distances"  when  the  first  sub-chord  is 
small. 

Caution.    It  will  not  be  satisfactory  in  this  case  to  produce 
the  curve  from  this  short  chord. 


75.   Problem. 


Given  D  and  stations  of  P.C.  and  P.T. 
Required  to  lay  out  the  curve  by  the  method 
of  Offsets  from  the  Tangent. 


Let  AG'  be  tangent  to  curve  AG 
Find 


AE'  =  c<  cos  «i 
EF"  =  100  cos  «2 
FG"  =  100  cos  <*3 


=  a3,  etc. 


EE'  =  Ci  sin  «i 
FF"  =  100  sin  «2 
GG"  =  100sina3 
FP  =  EE'  +  FF" 
GG'  =  FP  +  GG",  etc, 
When  AE  =  100,  then  \  d  becomes  \  D. 

76.  Field-work. 

(a)  Calculate  AE',   E'F,   FG' 

EE',   FP,     GG' 

(6)  Set  E',  P,  G'. 

(c)  Set  E  by  distance  AE  (cf)  and  EE'. 

(<f)  Set  F  "        "         EF  (100)  and  FP. 

(«)  Set  G  "        "         FG  (100)  and  GG'. 

For  the  computations  indicated  above,  always  use  natural 
sines  and  cosines. 


Simple  Curves. 


41 


77.   Ordinates. 

Problem.     Given  D  and  two  points  on  a  curve. 

Required   the    Middle    Ordinate    from    the 
chord  between  those  two  points. 


By  (9),  M= 

for  100  ft.  chord  M=  R  vers  £  D 

between  points  200  ft.  apart  M=  R  vers  D. 

Let  A  =  angle  at  center  between  the  two  points. 


78.   Problem.     Given  R  and  c. 
Required  M. 


Table  XXI.,  Allen,  gives  squares  and  square  roots  for  certain 
numbers.  If  the  numbers  to  be  squared  can  be  found  in  this 
'table,  use  (31).  Otherwise  use  logarithms  and  (32). 

79.   Problem.     Given  R  and  C. 

Required  the  Ordinate  at  any  given  point  Q. 

Measure  LQ  =  q.  Then  KN  =  Vjff2  -  g2 


-I)  (33) 


42 


Railroad  Curves  and  Earthwork. 


80.  When  C  =  100  ft.  or  less,  an 
approximate  formula  will  generally 
suffice. 

Problem.     Given  R  and  c. 

Required  M  (approx.) 

HL  :AH=^:J2 
AH2 


Where  AB  is  small  compared  with  J?, 
AH  =  -  (approx.) 


=        (approx.) 


(34) 


81.   Example.    Given  (7=100,  D  =  9°. 
Required  M. 

5730 


9 


Precise  value 
M=  1.965 


=  636.7 

8 

5093.6)  10000.  (1.963  = 
50936 
490640 
458424 


332160 
305616 


16544 

Table  XXVII. ,  Allen,  gives  middle  ordinates  for  curving  rails 
of  certain  lengths. 

82.    Problem.     Given  R  and  c. 

Required   Ordinate  at  any  given  point  Q. 

Approximate  Method. 
I.   Measure  LQ  =  q 


Jf=HL  = 
KK'  = 


(IT 

2R 
HK2 
2R 


Simple  Curves.  43 

Since    H  K  =  q  (approx.)  KK'  =-|?-J/  (approx.)    (35) 

\2/ 
KQ  =M-  KK' 

When  i=  1  as  in  figure,  KK'  =  ^  and  KQ  =  ?  Jf  (approx.) 

C       ^  44 


When  -  =  \  VW  =  ~M  (approx.  ) 

C        4  lO 

2 
When  -  =  T  TU  =  ^  If  (approx.) 

C        4  It) 

2 

The  curve  thus  found  is  accurately  a  parabola,  but  for  short 
distances  this  practically  coincides  with  a  circle. 

83.   II.  Approximate  Method.    Measure  LQ  and  QB 


(I)2-2  (£*«)(§-) 

KQ  =^n^  = o 

KQ=  AQ^QB  (approx.)  (36) 

&  M 

Sometimes  one,  sometimes  the  other  of  these  methods  will  be 
preferable. 

84.   Example.     Given  C  -  100,  D  =  9°. 

M-  1.965  from  Tables. 
Required,   Ordinate  at  point  30  ft.   distant 
from  center  toward  end  of  chord. 

I    30ft  -30x°  IL  AQ=     8° 

60     2  BQ^     20 

KK=^x    1.965  1273.4)1600(1.257 

9  1273.4 

25)17.685  -Ri  =  573°-         32660 

.70740  R9=    636.7       25468 

M=            1.965  2^9  =  1273.4         71920 

Ordinate  =          1.258  63670 

Precise  result  for  data  above  =  1.260.  8250 


Railroad  Curves  and  Earthwork. 


85.   Problem.     Given  R  and  c. 

Required  a  series  of  points  on  the  curve. 

H 


^  -p^k 


lf=HL=-^-(approx.) 

O  £l 


AH=£(approx.) 


PN  = 


pc 

—  (approx.),  etc.,  as  far  as  desirable. 
4 


This  method  is  useful  for  many  general  purposes,  for  ordi- 
nates  in  bending  rails  among  others. 

86.   Problem.     Given  a  Simple  Curve  joining  two  tangents. 
Required  the  P.  C.  of  a  new  curve  of  the  same 
radius  which  shall  end  in  a  parallel  tan- 
gent. 

Let   AB  be  the  given  curve. 

A'B'  "    "   required  curve. 

B'E  =  p  =  perpendicular  distance  between  tangents. 
Join  BB'. 

Then    AA'  =  00'  =  BB' 
Also  B'BE  =  V'VB  =  I 
BB'  sin  7  =  p 

BB'  =  AA'  = 


sin/ 


(37) 


When  the  proposed  tangent 
is  outside  the  original  tangent, 
the  distance  AA  is  to  be  added 
to  the  station  of  the  P.C.  When  inside,  it  is  to  be  subtracted. 


Simple  Curves. 


45 


87.    Problem.     Given  a  Simple  Curve  joining  two  tangents. 
Required  the  Radius  of  a  new  curve  which 
with  the  same  P.  C.  shall  end  in  a  parallel 
tangent. 

Let  AB  be  the  given  curve  of 
radius  R  =  AO. 

B'E  =  p  =  perpendicular 
distance. 

AB'  the  required  curve, 
radius  =  /?'. 

Draw  chords  AB,  AB' ; 
also  line          BB' ; 
also     BL  parallel  to  AO.' 
BL  =  00' 

=  B'L 
BLB'  =  AO'B' 


BLversBLB'  = 


-*>=db 


(38) 


Since  VAB  =  V'AB',  AB  and  AB'  are  in  the  same  straight  line. 
BB'=       2BL      sin^BLB' 

.R)sinJ/  (39) 


When  the  proposed  tangent  is  outside  the  original  tangent  (as 
it  is  shown  in  the  figure)  ,  the  above  formula  applies,  and 
R'>R. 

When  the  proposed  tangent  is  inside  the  original  tangent,  the 
formula  becomes 


R-  R'  = 


and  R<  <  R. 


vers/ 


(40) 


46 


Railroad  Curves  and  Earthwork. 


88.    Problem.     Given  a  Simple  Curve  joining  two  tangents. 
Required  the  radius  and  P.C.  of  a  new  curve 
to  end  in  a  parallel  tangent  with  the  new 
P.T.  directly  opposite  the  old  P.T. 

Let  AB  be  the  given  curve  of 
A  radius  =  It. 


A'B'  the  required  curve  of 
radius  R'. 

BB'=j>. 

Draw  perpendicular  O'N 

and  arc  NM 

Then  O'M  =  B'M  -  B'O' 

=  B'M  -  BM  =  BB' 
O'M  =p 


ON     exsec  NOO'  =  O'M 
-  R')  exsec  7     =p\ 


R-R'  = 


AA'  =  O'N  =  ON  tan  NOO' 
AA'  =  (R  -  .R')  tan    / 


exsec  / 


(41) 


(42) 


When  the  new  tangent  is  outside  the  original  tangent  (as  in 
the  figure),  R>R'  and  AA'  is  added  to  the  station  of  the  P.  C. 
When  the  new  tangent  is  inside  the  original  tangent,  R  <  72', 

RI  -  R  =  — P — ,  and  AA'  is  subtracted  from  station  of  P.  C. 
exsec  / 


89.  Problem.  To  find  the 
Simple  Curve  that  shall  join 
two  given  tangents  and  pass 
through  a  given  point. 

With  the  transit  at  V,  the 
given  point  K  can  often  be 
best  fixed  by  angle  BVK  and 
distance  VK.  If  the  point  K 
be  fixed  by  other  measure- 
ments, these  generally  can 
readily  be  reduced  to  the 
angle  BVK  and  distance  VK. 


Simple  Curves.  47 

90.   Problem.     Given  the  two  tangents  intersecting  at  F,  the 
angle  /,  and  the  point  K  fixed  by  angle 
BVK  =  /3  and  distance  VK  =  6. 
Required  the  radius  R  of  curve  to  join  the 
two  tangents  and  pass  through  K. 

In  the  triangle  VOK  \ve  have  given 

VK  =  b  and  OVK  =  18°  ~  7-  /S 

Further  VQ  =      E  OK  =  R 

cos  i/ 

VO  :OK  =  sinVKO  :  sin  OVK 

:  R  =  sin  VKO  :  cosQ  /+  0) 


cos 


From  data  thus  found,  the  triangle  VOK  may  be  solved  for  R. 

In  solving  this  triangle  the  angle  VOK  is  often  very  small.  A 
slight  error  in  the  value  of  this  small  angle  may  occasion  a 
large  error  in  the  value  of  R.  In  this  case  use  the  following 
Second  Method  of  finding  R  after  VOK  has  been  found. 


Find         AOK  =  f/  +  VOK        AlsoDVK  = 
Then  J?versAOK=LK 

=  b  sin  DVK 


E      b  sin  DVK 
"versAOK 


91.   Problem.     Given  R,  /,  0(BVK). 
Required  6(VK). 

In  the  triangle  VOK 

OK  =  ^:     OV= 


COS  ^  I 

OVK  =  90  -  (£  I  +  |8) 

Solve  triangle  for  b. 

Also  find  VOK  and  station  of  K  if  desired. 


48 


Railroad  Curves  and  Earthwork. 


92.  Problem.  To  find  the  point 
where  a  straight  line  intersects  a 
curve  between  stations. 

Find  where  the  straight  line  V'K 
cuts  VB  at  V. 
Measure  KV'B. 

Use  V  as  an  auxiliary  vertex. 
Find  /'  from  V'B. 
Solve  by  preceding  problem. 


93.   Approximate  Method. 

Set  the  middle  point  H  by  method  of  ordinates. 

If  the  arc  H  B  is  sensibly  a  straight  line,  find  the  intersection 
of  HB  and  CD. 

Otherwise  set  the  point  G  by  method  of  ordinates,  and  get 
intersection  of  HG  and  CD. 


Additional  points  on  the  arc  may  be  set  if  necessary,  and  the 
process  continued  until  the  required  precision  is  secured. 

The  points  H  and  G  can  be  set  without  the  use  of  a  transit 
with  sufficient  accuracy  for  many  purposes,  a  plumb  line  or  flag 
being  used  in  "sighting  in." 

94.   Problem.     Given  a  Simple  Curve  and  a  point  outside 

the  curve. 

Eequired  a  tangent  to  the  curve  from  that 
point. 

Let  BDE  be  the  given 
curve. 

P  the  point  out- 
side the  curve. 

BL  a  tan  gent  at  B. 
Measure  LBP,  also  B  P. 


Simple  Curves.  49 

In  the  triangle  BPO  we  have  given  PBO,  BP,  BO. 
Solve  the  triangle  for  BOP  and  OP. 

Then  cos  DOP  =§£  =  |, 

BOD  =  BOP  -  OOP 

From  BOD  find  station  of  D  from  known  point  B. 

It  should  be  noted  that  if  log  OP  is  found,  this  can  be  used 
again  without  looking  out  the  number  for  OP.  Other  similar 
cases  will  occur  elsewhere  in  calculation. 

When  for  any  reason  it  is  difficult  or  inconvenient  to  measure 
BP  directly,  the  angles  CBP,  BCP  and  the  distance  BC  may  be 
measured  and  BP  calculated. 

95.   Approximate  Method. 
Field-work. 

(a)  From  the  station  (B)  nearest  to  the  required  point  D, 
find  by  the  approximate  method  where  BP  cuts  the  curve  at  C. 
(If  E  be  the  nearest  station,  produce  PC  to  B.) 

(6)  Assume  D  with  BD  slightly  greater  than  CD,  and  with 
transit  at  P.  O.  set  the  point  D  (transit  point)  truly  on  the  curve. 

(c)  Move  the  transit  to  D,  and  lay  off  a  tangent  to  the  curve 
at  D.  This  will  very  nearly  strike  P. 


(d}  If  the  tangent  strikes  away  from  P,  at  Q,  measure  QDP, 
and  move  the  point  D  (ahead  or  back  as  the  case  may  be)  a  dis- 
tance c  due  to  an  angle  at  the  center  d  =  QDP.  The  tangent 
from  this  new  point  ought  to  strike  P  almost  exactly. 

In  a  large  number  of  cases  the  point  D  will  be  found  on  the 
first  attempt  sufficiently  close  for  the  required  purpose. 

If  a  tangent  between  two  curves  is  required,  similar  methods 
by  approximation  will  be  found  available. 


50 


Railroad  Curves  and  EartJiivork. 


Obstacles. 

When  any  obstacle  occurs  upon  a  u  tangent,"  the  ordinary 
methods  of  surveying  for  passing  such  obstacle  will  be  used. 

96.   When  V  is  inaccessible. 

Measure  VLM,  VML,  LM. 


LV  and  VM  are  readily  calculated, 
and  AL  and  MB  determined. 

In  some  cases  the  best  way  is  to 
assume  the  position  of  P.  C.  and  run 
out  the  curve  as  a  trial  line,  and 
finally  find  the  position  of  P.  C.  cor- 
rectly by  the  method  of  formula  (37) . 


97.   When  the  P.O.  is  inaccessible. 


Establish  some  point  D  (an  even 
station  is  preferable)  by  method  of 
"offsets  from  Tangent"  or  otherwise. 

Move  transit  to  B  (P.T.),  and  run 
out  curve  starting  from  D  and  checking 
on  tangent  VB. 


98.   When  the  P.  T.  is  inaccessible. 


With  instrument  still  at  F,  set  some 
convenient  point  D,  move  transit  to 
P.  C.,  and  run  in  curve  to  D,  and  then 
pass  the  obstacle  at  B  as  any  obstacle 
on  a  tangent  would  be  passed. 


Simple  Curves.  51 

99.  When  Obstacles  on  the  Curve  occur  so  as  to  prevent 
running  in  the  curve,  no  general  rules  can  well  be  given. 
Sometimes  resetting  the  transit  in  the  curve  will  serve.  Some- 
times, if  one  or  two  points  only  are  invisible  from  the  transit, 
these  can  be  set  by  "deflection  distances,"  and  the  curve  con- 
tinued by  "deflection  angles,"  without  resetting  the  transit. 
Sometimes  " offsets  from  the  tangent"  can  be  used  to  advan- 
tage. Sometimes  points  can  be  set  by  "  ordinates"  from 
chords.  Sometimes  the  method  shown  on  page  48,  §  92,  assum- 
ing an  auxiliary  F,  is  the  only  one  possible. 

It  should  be  borne  in  mind  that  it  is  seldom  necessary  that 
the  even  stations  should  be  set.  If  it  be  possible  to  set  any 
points  whose  stations  are  known  and  which  are  not  too  far 
apart,  this  is  generally  sufficient. 

Finally,  for  passing  obstacles  and  for  solving  many  problems 
which  occasionally  occur,  it  is  necessary  to  understand  the 
various  methods  of  laying  out  curves,  and  to  be  familiar  with 
the  mathematics  of  curves  ;  and,  in  addition,  to  exercise  a  rea- 
sonable amount  of  ingenuity  in  the  application  of  the  knowledge 


CHAPTEE  V. 
COMPOUND  CURVES. 

100.  When  one  curve  joins  another,  the  two  curves  having 
a  common  tangent  at  the  point  of  junction,  and  lying  upon  the 
same  side  of  the  common  tangent,  the  two  curves  form  a  Com- 
pound Curve. 

When  two  such  curves  lie  upon  opposite  sides  of  the  common 
tangent,  the  two  curves  then  form  a  Reversed  Curve. 

In  a  compound  curve,  the  point  at  the  common  tangent  where 
the  two  curves  join,  is  called  the  P.  C. (7.,  meaning  the  "point 
of  compound  curvature." 

In  a  reversed  curve,  the  point  where  the  curves  join  is  called 
the  P.E. Cy.,  meaning  the  "point  of  reversed  curvature." 

101.  Field-work. 

Laying  out  a  compound  curve  or  a  reversed  curve. 

(a)  Set  up  transit  at  P.  C. 

(&)  Run  in  simple  curve  to  P.  C.  C.  or  P.E.  C. 

(c)  Move  transit  to  P.  C.C.  or  P.E.  C. 

(d}  Set  line  of  sight  on  common  tangent  with  vernier  at  0. 

(e)  Run  out  second  curve  as  a  simple  curve. 

It  is  not  desirable  to  attempt  to  lay  out  the  second  curve 
with  the  transit  at  the  P.  C.  It  is  not  a  simple  and  convenient 
process  to  calculate  the  total  deflections  from  the  P.O.  to  a 
series  of  points  on  the  second  curve.  It  may  readily  be  shown 
that  adding  the  chord  deflection  for  the  second  curve  to  the 
total  deflection  for  the  P. C.C.  or  P.E.C.  will  yield  an  incor- 
rect result.  Resetting  at  the  P.  C.  C.  or  P.E. C.  is  quite  simple, 
and  the  process  of  running  in  the  second  curve  is  similar  in 
principle  to  that  of  §  59,  page  31. 

52 


Compound  Curves. 


53 


102.   Data  Used  in  Compound  Curve  Formulas. 

In  the  curve  of  larger  radius, 
v      L  ~A  _  -r. 


AOC  =  7j 
AV  =  Ti 
In  the  curve  of  shorter  radius, 


also 


BPC  =  7. 
VB=  T. 

LVB  =  7 


103.   Problem.     Given  BI,  7?,,  Ii,  /.. 
Required  7,  Ttj   Tt. 

Draw  the  common  tangent  DCE. 
Then      7=7z  +  7, 


EB  =  CE  = 

or  find  CD  and  CE  using  Allen's  Ta- 
ble II.  and  the  correction,  Table  IV. 
In  the  triangle  DVE  we  have  one 
side  and  three  angles 

72.  tan  ^  7.    (45) 


DE  =  J?, 
VDE  =  7, 
VED  =  7. 
DVE  =  180-7 


Solve  for  VD  and  VE 


AV  =  AD  +  VD  =  Tt 
VB  =  BE  -f  VE  =  Tt 

This  problem  will  be  of  use  in  making  calculations  for  platting, 
in  cases  where  the  curves  have  been  run  in  without  finding  V 
on  the  ground.  The  points  D  and  E  will  seldom  be  fixed  on  the 
ground,  and  in  cases  where  V  is  set,  the  problem  is  likely  to 
take  some  form  shown  in  one  of  the  problems  following. 


54 


Railroad  Curves  and  Earthwork. 


104.   Problem.     Given  T,,  /?,,  Si,  I. 
Required  Ttl  7j,  /.. 

A  L  V  U 


Draw  arcs  NP  and  KC. 

Draw  perpendiculars  MP,  LP,  SB,  UB. 

Then  AM  =  LP 

AN  =  R.  =  KP 

NM   =    LK  =       LS  KS 

OP  vers  NOP  =  VB  sin  VBS  -  PB  vers  KPB 
(Hi  —  7?a)  vers   7t     =  T»  sin    7—7?,  vers    / 


7,   =    I  ~   II 

AV=  MP  +.       SB  UV 

Tj  =  (Si  -  R*}  sin  Ii  +  B8  sin  /  -  Tg  cos  / 

105.   Problem.     Given  Ts,  Ks,  7g,  /. 
Hequired  Tt,  St,  It. 


(46) 

(47) 


(48) 


?  _  Tg  sin  I  —  E»  vers  7 

vers  7j 
rt  =  ( j£j  _  .Ra)  sin  7j  +  RS  sin  7  -  Tt  cos  7      (49) 


106.   Problem.     Given  Tj,  rg,  7?,,  7. 
Required  7?j,  7i,  7g. 

rgsin7-l?svers7 


Ti  +  T;  cos  7  -  BB  sin  7 

TI  +  Ts  cos  7  —  St  sin  7 

sin7t 


(50) 
(51) 


Compound  Curves. 


55 


107.   Problem. 
Given  Ti,Ei,  Rs,  I. 
Eequired  T.,  ItJ  /,. 

Draw  arcs  NP,  KC. 
Draw    perpendiculars    OK, 
AS,  PM,  VU. 

Then   LM  =  BP 
=  KN 

MN  =  LM  -  LN 
=  KN  -  LN 


LK  =  MN  =         KS  -        LS 

OP  vers  NOP  =  AO  vers  AOK  -  AV  sin  VAS 
-  Es}  vers    /,    =  EI  vers    I    —  TI  sin   / 
vers  I-  Tt  sin/ 


vers  I.  = 


—  Ea 


VB=      AS      -  PM  -     All 

Ts  =  EI  sin  I -(Ei-  7?s)  sin  /,  -  TI  cos  / 

108.   Problem.     Given        Tt,  EI,  It,  I. 
Eequired  Tt,  Es,  Ig. 


(53) 


vers  Ia 


109.  Problem.     Given        Tt.  Ts,  2tlt  7. 
Eequired  J?,,  /j,  /„. 

tan  i  7  = 


Tt 


(54) 


Tt  =  Et  sin  I -(Ei-  /?,)  sin  /.  -  Tt  cos  /       (55) 


(56) 


^-^  =  ^sin/-^cosj-^  (57) 

sin/. 


56 


Railroad   Curves  and  Earthwork. 


110.   Problem.     Given,  in  the  figure,  AB,  VAB,  VBA,  R,. 
Required  Hi,  Ii,  /,,  /. 

Draw  arc   NP ;    also  perpendicular 
KB,  MP,  SP. 

7=VAB  +  VBA 

NM  =        AK        +         KM         -  AN 
=  AB  sin  VAB  +  PBcosSPB  -  AN 
=  AB  sin  VAB  +  Rt  cos    I    —  R, 
=  AB  sin  VAB  -  It,  vers  / 
MP  =         KB        -         SB 
=  AB  cos  VAB-  PBsinSPB 
=  ABcosVAB-  .R.  sin   I 


OP  =  Ri  -  R,  = 


MP 
sin  L 


111.    Problem.     Given,  in  the  figure,  AB,  VAB,  VBA, 
Required  Rt,  It,  I,,  L 

Find  /,  and  show  that 

l  T  _  J?,  vers  7-  A  B  sin  VBA 
~J»»sin  J-ABcosVBA 

„       -„       Ri  sin  /-  AB  cos  VBA 
-ft  I  —  J+t  =:  -  :  —  7  — 
sm/, 


(58) 
(59) 


(60)    . 
(61) 


112.   Problem.     Given  a  Simple  Curve  ending  in  a  given 

tangent. 
A  second  curve  of  given  radius  is  to  leave  this  and  end  in 

a  parallel  tangent. 

Required  the  P.  C.  C. 

Let  AB  be  the  given  curve  of  ra- 

dius RI. 

CbetheP.C.a 
CB'  the  second  curve  of  ra- 

dius Rs. 
B  E  =  p  =  distance  between  tan- 

gents. 
Then  vers  COB  =  p  P  p        (62) 


Compound   Curves. 


57 


It  may  sometimes  be  more  convenient  or  quicker  to  run  in 
a  simple  curve  first  and  change  to  a  compound  curve  by  the 
method  of  this  problem,  rather  than  to  run  in  the  compound 
curve  at  first.  When  it  is  impossible  or  inconvenient  to  run  in 
the  curve  as  far  as  B  (P.  T.),  the  point  of  intersection  D  between 
the  curve  and  the  tangent  may  be  found,  the  angle  LDN  meas- 
ured, and  BE  calculated.  In  the  figure  below 

BE  =  DO  vers  DOB 


p= 


vers  LDN 


(63) 


113.  Example.     Given  Notes  of  Curve. 

22  +  20  P.O.       5°  curved 


Proposed  tangent  intersects 
curve  at  26  +  90. 

Angle  between  tangent  and 
curve  =  10°  20'. 

Required  Station  of  P.C.C.  to 
join  proposed  tangent,  using 
a  7°  curve. 

P  =  HB  vers  10°  20' 

E5    log  3.  059290 
10°  20'  vers  8.210028 


p  log  1.269318 


versCOB= 


26  +  90 
1  +  81.3 


08.7P.C'.C. 


Us  =  1146.28 
R7=    819.02 

327.26 

19° 24' 

10° 20' 

9°  04' 


p  log  1.269318 
log  2.514893 

vers  8. 754425 


9.0667(5° 
181.3 


58 


Railroad  Curves  and  Earthwork. 


114 

gent. 


.   Problem.     Given  a  Compound  Curve  ending  in  a  tan- 
Required  to  change  the  P.  C.  C.  so  as  to  end 
in  a  given  parallel  tangent,  the  radii  re- 
maining unchanged. 

I.  When  the  new  tangent  lies 
outside  the  old  tangent,  and  the 
curve  ends  with  curve  of  larger 
radius. 

Let  ACB  be  the  given  com- 
pound curve. 

AC'B'  the  required  curve. 
Produce  arc  AC  to  B". 
Draw  OB"  parallel  to  PB,  and 
--FT— [-;— }B'    B"E'  perpendicular  to  PB. 

Let  B'E  =  p  =  perpendicular 
distance  between  tangents. 
Then     B'E  =  B'E'  BE" 

B'E  =(7?f  -  7?8)  versC'P'B'-(7?j  -  7?,)  vers  COB" 
p    =  (Hi  -  7?,)  vers    7,'     -  (72,  -  72,)  vers    7, 
P 


vers  IJ  =  vers  Ii 
AOC' =  /-/,' 


(64) 


II.    When  the  new  tangent  lies  inside  the  old  tangent,  and 
the  curve  ends  with  the  curve  of  larger  radius. 


vers  7/  =  vers  Ii  — 


Ri-R. 


(65) 


III.   When  the  new  tangent  lies  outside  the  old  tangent,  and 
the  curve  ends  with  curve  of  smaller  radius. 


vers  /,'  =  vers  7,  — 


P 


7^-7?, 


(66) 


IV.   When  the  new  tangent  lies  inside  the  old  tangent,  and 
the  curve  ends  with  curve  of  smaller  radius. 


vers  I,'  =  vers  7.  -f 


-7?. 


(67) 


Compound  Curves. 


59 


115.    Problem.     Given  a  Simple  Curve  joining  two  tangents. 

Required  to  substitute  a  sym- 
metrical curve  with  flattened 
ends,  using  the  same  P.C. 
and  P.  T. 

Let  AHB  be  the  simple  curve  of 
radius  Re  . 

ACDB  the  required  curve  in 
which 

B=    AS    =  Rt 


ASC  =  BQD  =  7t 
CPD  =  7. 


7=  7.  +  27, 


PQO  =  7, 


There  are  then  Given  I,  Rc.     Required  7?,,  7?,,  7,,  7,. 

We  may  assume  any  two  of  the  latter  (except  7,  and  7,), 
and  readily  calculate  the  others. 

I.   Assume  7?,  and  7j. 

7.  =  7-  27, 

PQ      :     QO       =  sin  POQ  :  sin  OPQ 
7f,  -  7?,  :  RI  -  Rc  =  sin  £  7   :  sin  £  7. 


(68) 


II.   Assume  Rt  and  Rt. 


(69) 


60  Railroad  Curves  and  Earthwork. 

III.  Assume  Eg  and  7,. 


From  (68) 


Hi  sin  £  7,  - 
.Rj  sin  £  7  - 


3  sin  £  /g  =  EI  sin  £  7  -  Ec  sin  $  7. 
?j  sin  \  I,  =  7?c  sin  £  7  -  7?s  sin  £  7S. 
?c  sin  |  7  -  7?,  sin  |  7a 


sin  1  7-  sin  i  7, 


116.   Problem.     Given  a  Simple  Curve  joining  two  tangents. 
Required  to  substitute  a  curve  with  flattened 
ends  to  pass  through  the  same  middle 
point. 

Let  AB  be  the  given  simple  curve, 
and  H  the  middle  point. 

Erect  an  auxiliary  tangent  V'HV"  at  H. 

The  auxiliary  intersection  angles  at 
V  and  V"  are  readily  calculated  ;  also 
V'H  and  V"H. 

Sufficient  additional  data  can  be  as- 
sumed, and  the  problem  solved  as  a 
problem  in  compound  curves. 
It  is  not  necessary  that  the  curves  on  the  two  sides  of  H 
should  be  symmetrical. 


V'H  =  Tt 
VV'H  =  7 
7?{>AO 


Having  given 


Assume 


and  apply  (46)  and  (47). 

Other  assumptions  of  similar  sort  will  allow  the  use  of  other 
formulas  on  pages  54  and  55. 

Both  this  and  the  preceding  problem  will  be  found  of  con- 
siderable value  in  revising  the  alignment  of  track,  and  intro- 
ducing flatter  ends  for  the  curves  so  that  the  transition  from 
tangent  to  curve  shall  be  less  abrupt. 


Compound   Curves. 


61 


117.    Problem.     Given  two  simple  curves  with  connecting 

tangent. 

Eequired  to  substitute  a  simple  curve  of 
given  radius  to  connect  the  two. 

Let  DC  =  I  =  the  given  tangent,  connecting  the  two  curves 
AD  and  CB,  of  radii  Rs  and  7?j,  respectively. 


Let  EF  be  required  curve  of  radius  Re. 
Join  OP,  and  draw  perpendicular  OL. 


Then 


tanLOP  =  ^-  = 
OP  = 


cos  LOP 

In  the  triangle  OPQ  we  have  given 
I 


OP  = 


OQ  =  Sc  -  B, ;        QP  =  Ee  - 


cos  LOP' 

Solve  this  triangle  for  OQP,  QOP,  OPQ. 
Then  CPF  =  180°  -  (OPQ  +  OPL) 

EOD=    90°- (QOP  +  LOP) 


CHAPTER  VI. 


REVERSED  CURVES. 

118.  It  is  very  undesirable  that  reversed  curves  should  be 
used  on  main  lines,  or  where  trains  are  to  be  run  at  any  con- 
siderable speed.    The  marked  change  in  direction  is  objection- 
able, and  an  especial  difficulty  results  from  there  being  no 
opportunity  to  elevate  the  outer  rail  at  the  P.E.C.    The  use  of 
reversed  curves  on  lines  of  railroad  is  therefore  very  generally 
condemned  by  engineers.      For  yards  and  stations,  reversed 
curves  may  often  be  used  to  advantage,  also  for  street  rail- 
ways, and  perhaps  for  other  purposes. 

119.  Problem.     Given  the  perpendicular  distance  between 

parallel  tangents,  and  the  common  radius 
of  the  reversed  curve. 
Required  the  central  angle  of  each  curve. 

Let  AH  and  BD  be  the  par- 
allel tangents. 

ACB  the  reversed  curve. 
HB  =  p  =  perpendicular 
distance  between  tan- 
gents. 

Draw  perpendicular  NM. 
Let     AOC  =  BPC  =  Ir. 


Then 


120.   Problem. 


AO      PB       AO 


(71) 


Given  p,.  Ir. 
Required  It. 


vers/r 


(72) 


Reversed  Curves. 


63 


121. 


Problem.  Given  the  perpendicular  distance  between 
parallel  tangents,  and  chord  distance  be- 
tween P.  C.  and  P.  T. 

Eequired  the  common  radius  of  reversed 
curve  to  connect  the  parallel  tangents. 
P        Let  AH  and  BD  be  the  parallel 

tangents. 

ACB  the  reversed  curve. 
BH  =p 
AB  =  d 

Connect  AC  and  CB. 
Then  AOC  =  BPC,  and 

ACO  =  PCB 
ACB  is  a  straight  line 
AO:AL  =  AB:HB 


B 


,-J.-*.,, 


-: — 
4p 


122.   Problem. 


123.   Problem. 


Given  E  and  p. 
Eequired  d. 


(73) 


(74) 


Given  the  perpendicular  distance  between 
two  parallel  tangents,  and  the  central  an- 
gle and  radius  of  first  curve  of  reversed 
curve. 

Eequired  the  radius  of  second  curve. 
Let       ACB  =  reversed  curve 
HB=p 


AOC  =  CPB  =  Ir 

Draw  perpendicular  NCM. 
HB=         AN  +         MB 

=  AO  vers  AOC  +  BP  vers  BPC 
p  =  RI  vers  Ir    +  R2  vers  Ir 
P 


Vers  Ir 


(75) 


64 


Railroad  Curves  and  Earthwork. 


124.   Problem. 


Given  JBi,  J?2»  P- 
Required  Ir. 

vers/r  =  -=r^-, 


(76) 


125.   Problem.    Given  two  points  upon  tangents  not  parallel, 
the  length  of  line  joining  the  two  points,  and  the 
angles  made  by  this  line  with  each  tangent. 
Required  the  common  radius  of  a  reversed  curve 
to  connect  the  two  tangents  at  the  given  points. 

Let  A  and  B  be  the  given 
points. 

AL,  BM  =  given  tangents 
ACB  =  required  curve 
LAB  =  AandMBN  =  B 


Draw  PS  perpendicular 
and  OS  parallel  to  AB. 

Also  BD  perpendicular 
and  AD  parallel  to  OP. 

Then  PS  =         PU         +        SU 

OP  sin  POS  =  PB  cos  BPU  +  AO  sinOAB 
2J?sin  K   =  E  cos    B    +  E  cos  A 

.    7-^     cos  A  +  cosJ5 
sin  ±L  = 

AOK  =  0  =  180°  -  OKA  -      OAK 

=  180°-    K   -(90° -4)=  90°  +  A  -K 

=  P=180°-BKP-      PBK      = 

=  180°  -    K   -  (90°  -  5)  =  90°  +  B  -  K 
BD  =  AB  sin  DAB  =  AO  sin  AOK  +  BP  sin  KPB 
I  sin  K  =  E  sin    0    +  E  sin   P 
I  sin  K 


sin  O  +  sin  P 


(77) 


Reversed  Curves. 


65 


126.   Problem. 


Given  the  length  of  the  common  tangent  and 
the  angles  of  intersection  with  the  sepa- 
rated tangents. 

Required  the  common  radius  of  a  reversed 
curve  to  join  the  two  separated  tangents. 
Let  VAVB  =  common  tangent 
AVA,  BVB  =  separated  tangents 
ACB  =  required  curve 
LVAC  =  IA 
MVBB  =  IB 
VAVB  -  I 
VAVB  =      VAC 


VBC 


(78) 


An  approximate  method  is  as  follows  :  — 
Find  TAi  for  a  1°  curve  ;  also  Tsi  (Table  II.). 


Then 

127.   Problem. 


Da  = 


VAVB 


Given  a  P.O.  upon  one  of  two  tangents  not 
parallel,  also  the  tangent  distance  from 
P.  C.  to  V,  also  the  angle  of  intersection, 
also  the  unequal  radii  of  a  reversed  curve 
to  connect  the  tangents. 
Required  the  central  angles  of  the  simple 
curves,  and  tangent  distance,  V  to  P.  T. 
Let  AV  =  TI  =  given  tan- 
gent  distance. 
A  =  given  P.  C. 
V  =  vertex 
AVS  =  I 
AO  = 


VS  =  second  tangent 
ACB  =  required  curve 
AOC  = 
BPC  = 
BV  =  TZ  =  required  tan- 
gent distance. 


66 


Railroad  Curves  and  Earthwork. 


Draw  arcs  OU,  ON,  also  perpendiculars  AN,  OM,  NL. 
Then  AO  =  BU  =  J?i 

LU  =  BU  -    BL  =  AN-SN=AS 

LU  =  AS  MU  -         ML 

AVsinAVS=        PO       vers  BPC  -  AO  vers  OAN 
Ji  sin  I    =  (Ei  +  jft2)  vers    72    -  RI  vers    / 


vers  72  = 


sin  7 


(79) 


/I  =  /2  -  / 

Tz  =  T!  cos  7  +  RI  sin  7  -  (7?i  +  7?2)  sin  72  (80) 

128.   Problem.     Given  BV  instead  of  AV,  and  other  data  as 

above. 
Required  /i,  72,  etc. 

Draw  a  new  figure  similar  in  principle  to  the  preceding  and 
solve  in  a  similar  way,  and  using  the  same  notation  as  above 

yers/^^  +  J251117  (81) 

Ti  =  Tz  cos  /  +  Ez  sin  I  +  (Ri  +  Rz)  sin  /i        (82) 


CHAPTER   VII. 
PARABOLIC  CURVES. 

129.  Instead  of  circular  arcs  to  join  two  tangents,  parabolic 
arcs  have  been  proposed  and  used,  in  order  to  do  away  with 
the  sudden  changes  in  direction  which  occur  where  a  circular 
curve  leaves  or  joins  a  tangent.     Parabolic  curves  have,  how- 
ever, failed  to  meet  with  favor  for  railroad  curves  for  several 
reasons. 

1.  Parabolic  curves  are  less  readily  laid  out  by  instrument 
than  are  circular  curves. 

2.  It  is  not  easy  to  compute  at  any  given  point  the  radius  of 
curvature  for  a  parabolic  curve  ;  it  may  be  necessary  to  do  this 
either  for  curving  rails  or  for  determining  the  proper  elevation 
for  the  outer  rail. 

3.  The  use  of  the  "  Spiral,"  or  other  "  Easement,"  or  "  Tran- 
sition "  curves  secures  the  desired  result  in  a  more  satisfactory 
way. 

There  are  however  many  cases  (in  Landscape  Gardening 
or  elsewhere)  where  a  parabolic  curve  may  be  useful  either 
because  it  is  more  graceful  or  because,  without  instrument,  it  is 
more  easily  laid  out,  or  for  some  other  reason. 

It  is  seldom  that  parabolic  curves  would  be  laid  out  by 
instrument. 

• 

130.  Properties  of  the  Parabola. 

(«)  The  locus  of  the  middle  points  of  a  system  of  parallel 
chords  of  a  parabola  is  a  straight  line  parallel  to  the  axis  of  the 
parabola  (i.e.  a  diameter). 

(6)  The  locus  of  the  intersection  of  pairs  of  tangents  is  in 
the  diameter. 

(c)  The  tangent  to  the  parabola  at  the  vertex  of  the  diameter 
is  parallel  to  the  chord  bisected  by  this  diameter. 

(d)  Diameters  are  parallel  to  the  axis. 

67 


68 


Railroad  Curves  and  Earthwork. 


(c)  The  equation  of  the  parabola,  the  coordinates  measured 
upon  the  diameter  and  the  tangent  at  the  end  of  the  diameter  is 

.,*•_   4^  w 


— 

sin20 


or 


(83) 

131.   Problem.     Given  two  tangents  to  a  parabola,  also  the 

position  of  P.O.  and  P.  T. 
Required  to  lay  out  the  parabola  by  **  off- 
\x  sets  from  the  tangent." 


Let  AV,  VB  be  the  given  ^-^  \ 

tangents  (not  necessarily  equal),  ^-^  \ 

and  AHB  the  parabolic  curve.  "^\^  \ 

Join  the  chord  AB.  "*-^Y 

Draw  VG  bisecting  AB. 

Draw  AX,  BY,  parallel  to  VG. 

Produce  AV  to  Y. 

Then  VG  is  a  diameter  of  the  parabola. 
AX  parallel  to  VG  is  also  a  diameter. 

The  equation  of  the  parabola  referred  to  AX  and  AY  as  axes  is 

t/2  =  4p'x. 

Hence  AV2  :     AY2     =  HV  :  BY* 

AV2  :  (2  AV)2  =  HV  :  2  GV 


=HV:2GV 


HV=f 


(84) 


Next  bisect  VB  at  D. 
Draw  CD  parallel  to  AX. 


Then 


BD2  :  BV2  =  CD  :  HV 


Parabolic  Curves.  69 


Similarly,  make  AN  =  NF  =  FV 

Then  KN  =  — 


In  a  similar  way,  as  many  points  as  are  needed  may  be 
found. 

132.   Field-work. 

(a)  Find  G  bisecting  AB. 
(6)  Find  H  bisecting  GV. 

(c)  Find  points  P,  Q,  and  N,  F,  dividing  AG,  AV,  proportion- 
ately ;  also  R  and  D,  dividing  GB  and  BV  proportionately. 

Use  simple  ratios  when  possible  (as  £,  £,  etc.). 

(d)  Lay  off  on  PN,  the  calculated  distance  KN 

on  QF  lay  off  EF 
on  RD  lay  off  CD 

In  figure  opposite,  KN  =  — 

9  , 

CD=T 
EF=|HV 

For  many  purposes,  or  in  many  cases,  it  will  give  results 
sufficiently  close,  to  proceed  without  establishing  P,  Q,  R  ;  the 
directions  of  NK,  EF,  CD,  being  given  approximately  by  eye. 
When  the  angle  AVG  is  small  (as  in  the  figure),  it  will  generally 
be  necessary  to  find  P,  Q,  R,  and  fix  the  directions  in  which 
to  measure  NK,  EF,  CD.  When  the  angle  AVG  is  large  (greater 
than  60°)  and  the  distances  NK,  EF,  CD  are  not  large,  it  will 
often  be  unnecessary  to  do  this.  No  fixed  rule  can  be  given 
as  to  when  approximate  methods  shall  be  used.  Experience 
educates  the  judgment  so  that  each  case  is  settled  upon  its 
merits. 


70 


Railroad  Curves  and  Earthwork. 


133.   Problem.     Given  two  tangents  to  a  parabola,  also  the 

positions  of  P.  C.  and  P.  T. 
Required  to  lay  out  the  parabola  by  "mid- 
dle ordinates." 


T  „-• 


The  ordinates  are  taken  from  the  middle  of  the  chord,  and 
parallel  to  GV  in  all  cases. 

Field-work. 

(a)  Establish  H  as  in  last  problem. 

(6)  Lay  off  SE  =  \  HV  ;  also  TC  =  ±  HV. 

(c)  Lay  off  UW  =  ^  TC,  and  continue  thus  until  a  sufficient 
number  of  points  is  obtained. 

The  length  of  curve  can  be  conveniently  found  only  by  meas- 
urement on  the  ground. 
,  Note  the  difference  in  method  between  8  85  and  §  133. 


134.  Vertical  Curves. 

It  is  convenient  and  customary  to  fix  the  grade  line  upon  the 
profile  as  a  succession  of  straight  lines  ;  also  to  mark  the  eleva- 
tion above  datum  plane  of  each  point  where  a  change  of  grade 
occurs ;  also  to  mark  the  rates  of  grade  in  feet  per  station  of 
100  feet.  At  each  change  of  grade  a  vertical  angle  is  formed. 
To  avoid  a  sudden  change  of  direction  it  is  customary  to  intro- 
duce a  vertical  curve  at  every  such  point  where  the  angle  is 
large  enough  to  warrant  it.  The  curve  commonly  used  for  this 
purpose  is  the  parabola.  A  circle  and  a  parabola  would  sub- 
stantially coincide  where  used  for  vertical  curves.  The  parabola 
effects  the  transition  rather  better  theoretically  than  the  circle, 
but  its  selection  for  the  purpose  is  due  principally  to  its  greater 
simplicity  of  application.  It  is  generally  laid  to  extend  an 
equal  number  of  stations  on  each  side  of  the  vertex. 


Parabolic   Curves. 


71 


135.   Problem.     Given  the  elevations  at  the  vertex  and  at  one 

station  (100'}  each  side  of  vertex. 
Required  the  elevation  of  the  vertical  curve 
opposite  the  vertex. 


Let  A,  V,  B,  be  the  given  points, 

AHB  the  parabola. 
Join  AB. 

Draw  vertical  lines  AX,  LGHV,  MBY,  and  horizontal  line  ALM. 
Produce  AV  to  Y. 


In  the  case  of  a  vertical  curve,  the  horizontal  projections  of 
AV  and  VB  are  equal,  and  here  each  equals  100  feet  =  AL  =  ML 


Therefore 


AG  =  GB,  and  AV  =  VY 


VG  is  a  diameter  of  the  parabola. 
AX  is  also  a  diameter. 


Elev.  H  = 


Elev. 


(85) 


This  affords  a  simple  and  quick  method  of  finding  H  when 
the  vertical  curve  extends  only  one  station  each  side  of  vertex, 
which  is  the  most  common  case.  Other  methods  or  rules  for 
vertical  curves  are  used  on  various  railroads,  or  by  different 
engineers,  and  it  will  prove  interesting  and  valuable  to  investi- 
gate such  rules,  to  discover  whether  the  resulting  curve  is  a 
parabola,  as  will  generally  be  found  to  be  the  case.  When  the 
vertical  curve  extends  more  than  one  station  each  side  of  the 
vertex,  the  following  method  is  preferable,  which  is  also  appli- 
cable to  the  above  case,  and  is  in  some  respects  preferable  for 
that  also. 


72 


Railroad  Curves  and  Earthwork. 


136.   Problem.     Given  the  rates  of  grade  g  of  AV  ;  g'  of  VB  ; 
the  number  of  stations  n,  on  each  side  of 
vertex,  covered  by  the  vertical  curve ;  also 
the  elevation  of  the  point  A. 
Required  the  elevation,  at  each 
station,  of  the  parabola  AB. 

Draw  vertical  lines 

DD'D",  EE'E",  VHL,  YBM 
Also  horizontal  lines 
VC,  ALM 

Produce  AV  to  Y 

-    >^» 

I 

i 

~D""~~     ~\T'  L" 


---iC 


Let  a\  —  offset  DD'  at  the  first  station  from  A. 
a2  =  "  EE'  "  second  "  "  A. 
«3  =etc. 

Then       «2  =     2%i     =  4  a\ 
'  az  =     32ai     =  9  ai 
a2n  =  (2  n)2ai  =  4  w2ai  =  YB 
YB  =      YC  -  BC 
4  n2ai  =      ng  -  ng' 

ai=sf£ 


(86) 


Due  regard  must  be  given  to  the  signs  of  g  and  g'  in  this 
formula,  whether  +  or  — . 


Parabolic  Curves.  73 

From  the  elevation  at  A  we  may  now  find  the  required  eleva- 
tions, since  we  have  given  0, 

and  we  also  have      a\ 

a2  =  4  ai 
a$  =  9  ai  etc. 

A  method  better  and  more  convenient  for  use  is  given  below. 

DD"=     0;  D'D"  =     g  -  ai 

EE"  =  2  g  ;  E'E"  =  2g-a2  =  2g-    4  ai 

VL     =  30;  HL     =  30-a3  =  30-9ai 

RT    =  4  g  ;  TS     =  4  0  -  a*  =  4  0  -  16  ai  etc. 

Again,  D'D"  =    g  -      ai  =  g  -    ai 

E'E"  -  D'D"  =  2  g  -  4ai-(  g-  ai)=0-3ai 
HL  -E'E"  =3gr-  9ai-(20-4ai)  =  0-5ai 
ST  -  HL  =  40-16ai-(30-9ai)=0-7ai  etc. 

On  a  straight  grade,  the  elevation  of  any  station  is  found 
from  the  preceding,  by  adding  a  constant  g. 

On  a  vertical  curve,  the  elevation  of  each  station  is  found 
from  the  preceding  by  adding,  in  a  similar  way,  not  a  constant, 
but  a  varying  increment,  being  for  the 


1st  station  from  A  =  0  —    a\ 
2d       "        "      A  =  <7-3«i 


3d 


changing  by  successive 
differences  of  2  ai  in 
each  case. 


The  labor  involved  is  not  materially  greater  in  many  cases, 
for  a  vertical  curve  than  for  a  straight  grade.  This  method  has 
the  additional  advantage  that  a  correct  final  result  at  the  end 
of  the  vertical  curve  makes  a  "check"  upon  all  intermediate 

results. 


74  Railroad  Curves  and  Earthwork. 

137.   Example. 


Given.     Grades  as  follows 

:—              Sta. 

Elev. 

Rate 

5 

1  17.50 

Then  a\  = 

g  -  g>  _  3.00  _ 
4n    "  12 

=  0.25             10 

15 

135.00 
137.50 

+  3.50 
+  0.50 

Sta. 

Elev. 

n  =  3.50  - 

0.50  =  3.00 

(88). 

5 

117.50 

+  3.50 

-9 

6 

121.00 

+  3.50 

3.50 

=  9 

7 

124.50 

-0.25 

=  ai 

+  3.25 

3.25 

=  ff—     GI 

8 

127.75 

-0.50 

-2ai 

+  2.75 

2.75 

g  -3ai 

9 

130.50 

-0.50 

-2ai 

+  2.25 

2.25 

g  -5ai 

etc. 

10 

132.75 

-0.50 

+  1.75 

1.75 

11 

134.50 

-0.50 

+  1.25 

1.25 

12 

135.75 

-  0.50 

+  0.75 

0.75 

13 

136.50 

End  of  curve 

+  0.50 

=  9' 

14 

137.00 

+  0.50 

15 

137.50 

The  elevation  for  Sta.  15  thus  obtained  agrees  with  the  eleva- 
tion shown  in  the  data.  All  the  intermediate  elevations  are 
therefore  "checked." 

138.   Problem.     Given    g,  g',  a\ 
Required  n 

From  (86)  n=2_=LSL'  (87) 

From  practical  considerations  ai  should  not  exceed  0.25  or 

n^^^rOTn=g-g'  (88) 


CHAPTEK  VIII. 
TURNOUTS. 

139.   A  Turnout  is  a  track  leading  from  a  main  or  other 
track. 
Turnouts  may  be  for  several  purposes. 

I.    Branch  Track  (for  line  used  as  a  Branch  Road  for  gen- 
eral traffic). 

II.    Siding  (for  passing  trains  at  stations,  storing  cars, 

loading  or  unloading,  and  various  pur- 


III.  Spur  Track       (for  purposes  other  than  general  traffic, 

as  to  a  quarry  or  warehouse). 

IV.  Cross  Over        (for  passing  from  one  track  to  another, 

generally  parallel). 

The  essential  parts  of  a  turnout  are 

1.    The  Switch.          2.    The  Frog.          3.    The  Guard  Bail. 

1.  Some  device  is  necessary  to  cause  a  train  to  turn  from  the 
main  track  ;  this  is  called  the  "  Switch." 

2.  Again,  it  is  necessary  that  one  rail  of  the  turnout  track 
should  cross  one  rail  of  the  main  track ;  and  some  device  is 
necessary  to  allow  the  flange  of  the  wheel  to  pass  this  crossing  ; 
this  device  is  called  a  "  Frog." 

3.  Finally,  if  the  flange  of  the  wheel  were  allowed  to  bear 
against  the  point  of  the  frog,  there  is  danger  that  the  wheel 
might  accidentally  be  turned  to  the  wrong  side  of  the  frog 
point.     Therefore  a  Guard  Hail  is  set  opposite  to  the  frog,  and 
this  prevents  the  flange  from  bearing  against  the  frog  point. 


76 


Railroad  Curves  and  Earthwork. 


Frogs  are  of  various  forms  and  makes,  but  are  mostly  of  this 
general  shape,  and  the  parts  are  named  as  follows  :  — 


M  =  mouth 
O  =  throat 

WW  =  wings 


P  =  point 
T  =  tongue 
H  =  heel 


This  shows  the  « stiff"  frog. 

The  "spring"  frog  is  often  used  where  the  traffic  on  the 
main  line  is  large,  and  on  the  turnout  small.  In  the  spring 
frog  WW  is  movable.  AD  represents  the  main  line,  and  W'W' 


is  pushed  aside  by  the  wheels  of  a  train  passing  over  the  turn- 
out. The  "Frog  angle"  is  the  angle  between  the  sides  of  the 
tongue  of  the  frog  =  APB. 

Frogs  are  made  of  certain  standard  proportions,  and  are  clas- 
sified by  their  number. 

The  "Number"  n  of  a  frog  is  found  by  dividing  the  length 

PH 
of  the  tongue  by  the  width  of  the  heel ;  that  is,  n  = 

»  140.    Problem.     Given  n.    Required  Frog  Angle  F. 


(89) 


Turnouts.  77 

141.  A  form  of  switch  in  common  use  is  the  "  Stub- Switch," 
which,  is  formed  by  two  rails,  one  on  each  side  of  the  track, 
called  the  Switch  Bails.     One  end  of  the  rail  for  a  short  dis- 
tance (often  about  5  feet)  is  securely  spiked  to  the  ties,  the 
rest  of  the  rail  being  free  to  slide  on  the  ties,  so  that  it  may 
meet  the  fixed  rails  of  either  main  track  or  turnout,  as  desired. 
These  fixed  rails,  supported  on  a  Head  Block,  are  held  by  a 
casting,  or  piece  of  metal  called  the  Head  Chair,  and  upon 
which  the  switch  rail  slides.     A  Switch  Eod  connects  the  ends 
of  the  switch  rails  with  the  Switch  Stand.     One  end  of  the  rail 
is  spiked  down,  so  that  when  the  free  end  is  drawn  over  by  the 
switch  rod,  the  rail  is  sprung  into  a  curve  which  may  with 
slight  error  be  considered  a  circular  curve,  tangent  to  the  main 
line  (if  this  be  straight).     The  distance  through  which  the  free 
end  of  the  rail  is  drawn  or  Thrown  by  the  switch  rod  is  called 
the  Throw  of  the  switch.     The  free  end  of  the  rail  is  called  the 
Toe,  and  the  P.  C.  of  the  curve  the  Heel  of  the  switch. 

Knowing  the  throw  t  and  the  length  I  of  the  switch  rail,  we 
can  deduce  the  radius  JR,  or  degree  of  curve  D,  and  continuing 
this  curve  to  the  point  of  frog,  we  can  readily  deduce  the  angle 
between  the  rails  or  the  Frog  Angle  necessary. 

It  is  more  customary,  however,  having  given  the  throw 
(5"  _  5^'  _  6"  are  used  on  different  roads)  to  assume  either 

(1)  the  radius  (or  degree)  of  turnout  curve,  and  from  this 
find  F  (or  n)  and  I ;  or 

(2)  the  number  n  (or  angle  F}  of  frog,  and  from  this  find  E 
and  then  I.    Table  XXII.  gives  such  data. 

142.  When  there  are  two  turnouts  at  the  same  point,  one  on 
each  side  of  the  main  line,  three  frogs  are  necessary,  the  middle 
one  being  called  the  "  Crotch  Frog." 

It  is  necessary  that  there  should  be  two  numbers  of  frog,  one 
for  the  ordinary  turnout  frog  and  another  for  the  "crotch 
frog."  It  is  advisable  that  only  two  be  used,  and  that  all  turn- 
out curves  be  arranged  to  use  one  or  the  other  of  these  two 
frogs.  (For  double  turnouts  with  point  switches  a  third  num- 
ber may  be  necessary  for  a  crotch  frog.  In  this  case  we  should 
assume  n,  or  F,  as  the  value  for  one  of  the  two  standard  frogs.} 

On  main  line  it  is  now  customary  to  use  a  "  Split  Switch  "  or 
"  Point  Switch,"  the  description  and  discussion  of  which  will 
follow  the  discussion  of  the  stub  switch. 


78 


Railroad  Curves  and  Earthwork. 


In  the  figure  the 

Heel  of  Switch  is  at  E  or  Q 
Toe  of  Switch 
Head  Block 
Crotch  Frog 


Length  of  Switch  Rail  =  QH 
H  Throw  of  Switch  =  HI 

H  and  I     Lead  =  Fl 

C  Frog  is  at  F 


Center  of  turnout  curve   AB 


Q     A    E 


143.   Problem.     Given  gauge  of  track,  g,  frog  angle,  F,  and 

throw  of  switch,  t. 
Required  ft,  I,  and  QF  =  E. 

Let  EF,  QR,  be  the  rails  of  the  turnout. 

Draw  perpendicular  KF.     Also  HI  at  toe  of  switch. 


QF  =      OF     sin  EOF 

E  =  (.R  +  ^sm    F 
(92) 

by  (26)  t  =  •—    (approx.) 

I  =  V2Rt  (approx.) 
(93) 


Turnouts.  79 

144.    Problem.     Given  g,  t,  n. 

Required  R,  E,  I. 

In  the  figure  preceding  connect  EF. 
Then  EFQ  =  FEK  =  1 F 

QF  =  EQ  cot  EFQ 

E  =  g  cot^F  (94) 

=  2  ng  (95) 

FQ2=      FO2      -     QO2 

^2 


E2=  2R  x      g 


2flr       2gr 
=  2  ?i2gr  (96) 


(97) 


145.   Problem.     Cri'ven  fir. 

Required  the  middle  ordinate  ST. 

from  (34)        *f  =  ^ 
o  xt 

EQ  is  middle  ordinate  for  chord  2  FQ 

SJ     U  4(  it  44  U  £p 

EF  =  FQ  (approx.) 
ST:EQ=EF2:(2FQ)2 


This  is  true  evidently  whatever  the  degree  of  curve. 


80 


Railroad  Curves  and  Earthwork. 


146.   Problem.     Given  g,  R. 
Required,  the  angle  of 
crotch  frog,  C. 


The  frog  angle  at  C  =  2  AOC 


147.   Problem.     Owen  the  number  of  crossing  frog    —  nf. 
Required  the  number  of  crotch  frog  =  nc. 

AO  =  R  =  2  nfg 

If  we  consider  AD  to  represent  a  rail  and  nq  the  frog  proper 
for  the  crossing  of  QC  and  AD, 


the 


UO  =  2  n*  g- 
2 


But  the  angle  between  EC  and  QC  =  twice  the  angle  between 
QC  and  AD. 


Then 


^     (approx.) 


(approx.) 


w/2  =      2  nc2 
0.7071n/=  nc 


(approx.) 
(approx.) 
(approx.) 


(100) 


The  approximation  indicated  here  gives  results  much  more 
nearly  correct  than  it  is  practicable  to  work  to  in  practice. 
For  instance,  in  the  case  of  a  No.  9  frog,  the  number  of  the 
corresponding  crotch  frog  is  6.359  by  Table  XXII. 

By  (100)  it  is  6.364 

Railroads  use  6.5 


Turnouts. 


81 


148.  Problem.     Given  main    E 
track  of  radius  Em;  also  F,  g, 
and  n. 

Eequired  radius  Et  of  a  turn- 
out inside  of  main  track  ;  alsoE. 

Let  EB  be  the  outer  rail  of 
main  track. 

EF  the  outer  rail  of  turnout. 
Join  EF. 

Let     EOF  =  0;   PFO  =  F. 
In  the  triangle  EOF,  we  have 

EO  =  Km 


EFO  -  FEO  =  EFO  -  EFP  =  F 
EFO  +  FEO  =  180°  -  0 

Then  tan  KEFO  + FEO):  tan  £(EFO- FEO)  =  EO-fFO:Eflf-FO 
cot  £  0  :  tan  £  F  =    2  Em    :   g 

tan  £  0  :  cot  \  F  =       g       : 


(101) 


tan*0=2-kcot^=^- 

tan}0=£ 

Similarly,  FPH  = 

Join  HF,  and  in  triangle  HPF 


gn 


0) 


chord  HF  =  E  -  2  (ltm  -  1^  sin  \  O 


(102) 


(103) 


82  Railroad  Curves  and  Earthwork. 

149.   Approximate  Formula. 

Let  B,  D  —  Radius  and  a  Degree  of  a  turnout  curve  from  a 
straight  line  to  correspond  to  the  given  value 
of  F  or  n. 

J?m,  Dm  =  Radius  and  Degree  of  main  track. 
jR«,  Dt  =  Radius  and  Degree  of  turnout  curve. 

Then  from  (101)  (102)  Em  =     ng     ;  Rt  = 

' 


also  (96)  E  =  2  n*g  =  ng  x  2  n  = 


ng 


sin 


t  ng 

50      50  tan    F 


E  ng 

sin  £  j[)w  :  sin  |  JDt  :  sin  £  Z>  =  tan  $  0  :  tan  $(  0+  .F)  :  tan  \  F 

Dm\          Dt  :          D=  O:       0  +  F       .-^(approx.) 

Dm  +  D         :          Dt=  0+  F  :  0  +  F        (approx.) 

Dt  =  Dm  +  D  (approx.)  (104) 

Again,      (103)  HF  =  E  =  2  (ltm  -  f\  sin  \  O 

But  ^  is  small  compared  with  Hm,  and  may  be  neglected, 
and  for  small  angles  sin  £  0  =  tan  \  0  (approx.) 

HF  =  E  —  2  Em  tan  £  O  (approx.) 
(101)  E  =  2gn  (approx.  X  (105) 

This  agrees  with  (95)  E=2ng  (turnout  from  straight  track) 

The  above  formula  and  (104),  while  approximate,  are  the 
formulas  in  general  use. 

It  is  difficult  in  practical  track  work  to  secure  results  more 
precise  than  would  be  obtained  by  tl.e  use  of  these  approximate 
formulas. 


Turnouts. 


83 


150.   Problem.     Given  main  track  of  radius  Jf?m,  also  F,  g,  n. 
Required  radius  Et  of  a  turnout  curve  out- 
side of  main  track. 


I.  When  the  center  of  turnout  curve  lies 
outside  of  main  track. 

Let  EB  be  the  outer  rail  of  main  track, 
and  H  F  the  outer  rail  of  turnout. 

Join  HF. 

Let  HOF  =  O 

Then  PFL  =  F 

In  the  triangle  HOF 

FO  ==£„  +  {[ 


Also     FHO+HFO  =  180°-  0 

FHO  -  HFO  =  180°  -  PHF  -(180°  -  PHF  -  F) 

=  F 
Then 

tan£(FHO  +  HFO):  tan|(FHO  -  HFO)=  FO  +  HO  :  FO  -HO 
cot  I  0  :  tan  i  F  =     2  Mm     j    g 

tan  \  0  =  —  (106) 

Similarly,  OPF  =  F  -  O 
Join  EF,  and  in  triangle  EPF 


gn 


tan|(J*~  O) 

chord  EF  =  E  =  2    fim  + 


Approximate  Formulas. 

Dt  -  D  -  Dm  (approx.) 
E  =  2ng  (approx.) 


(107) 
(108) 

(109) 


84  Railroad  Curves  and  Earthwork. 

151.   II.  When  the  center  of  turnout  curve  lies  on  the  inside 
of  main  track. 
By  a  process  entirely  similar  it  may  be  shown  that 


(110) 


=  2    .»„  +       sin  JO  (112) 


Approximate  Formulas. 

Dt  =  Dm-D  (approx.)  (113) 

E  =  2ng         (approx.) 

152.   Example.     Given  a  3°  curve  on  main  line  and  a  No.  9 

frog. 

Required  the  degree  of  turnout  curve  to  the 
inside  of  the  curve. 

Table  XI,  Searles',  shows  for  a  No.  9  frog  the 

degree  of  curve  =    7°  31'  04"  ;  this  is  ordinarily 

taken  =    7°  30'       =  D 

degree  of  main  line     =    3°  00'        =  Dm 

degree  of  turnout        =  10°  30'        =  Dt  =  D  +  Dm 

By  precise  formula  10°  32'        =  Dt 

The  difference  between  approximate  and  precise  results  is 
commonly  small  enough  to  have  little  effect  in  the  short  dis- 
tance to  the  point  of  frog. 

In  a  similar  way  for  a  turnout  on  the  outside  of  the  main, 
line,  using  a  No.  9  frog,  the  degree  of  curve  would  be 
D  -  Dm  =  4°  30' 


Turnouts.  85 

153.    Split  Switch. 

The  stub  switch  fails  to  meet  the  requirements  of  modern 
railroad  practice.  At  the  head  block,  the  sliding  end  of  the 
switch  rail  is  not  held  firmly  down  to  the  ties.  Rails  are  also 
found  to  "  creep"  or  travel  longitudinally,  so  that  the  joint  at 
the  head  block  is  often  too  wide  and  again  is  liable  to  close  up  so 
tight  that  the  switch  rail  cannot  be  moved. 

In  turnouts  from  main  line,  it  is  customary  now  to  use  the 
split  switch.  In  this  switch  the  outer  rail  of  the  main  line  and 
the  inner  rail  of  the  turnout  track  are  continuous. 

The  switch  rails  are  steel  rails,  each  planed  down  at  one  end 
to  a  wedge  point,  so  that  it  may  be  caused  to  lie  close  against 
the  track  rail,  and  so  turn  the  wheel  in  the  direction  intended. 
An  angle  is  made  between  the  main  track  and  the  switch  rail. 
The  fixed  end  of  the  switch  rail  is  placed  at  a  point  correspond- 
ing to  the  head  block  of  the  stub  switch,  and  the  distance 
between  rails  at  this  point  is  generally  made  the  same  as  the 
"throw"  of  the  stub  switch  (from  5"  to  6").  The  switch  rail 
is  often  made  15  feet  in  length,  but  a  length  of  19  feet  is  not 
uncommon.  The  "switch  angle  "  is  determined  by  the  length 
of  switch  rail  and  this  distance  between  rails  (gauge  to  gauge), 
and  which  we  may  call  t. 

With  the  split  switch  a  common  practice  is  to  calculate  the 
turnout  for  a  stub  switch ;  the  fixed  end  of  the  split  switch  is  then 
placed  at  the  point  where  the  movable  end  of  a  stub  switch 
would  be  placed,  and  the  point  of  the  switch  wherever  this  will 
bring  it.  This  gives  results  approximately  correct,  and  suffi- 
ciently good  to  satisfy  the  requirements  of  many  railroads. 
For  instance  if  I  =  25  feet,  and  the  length  of  switch  rail  is  15 
feet,  the  P.O.  of  curve  will  come- 10  feet  back  from  the  point  of 
switch,  and  the  P.  C.  of  the  center  line  of  turnout  curve  will  be 
on  the  center  line  of  main  track.  The  formulas  derived  for  the 
use  of  stub  switches  are  available  therefore  in  many  cases  where 
split  switches  are  used.  Some  prefer  a  more  exact  solution.  If 
the  solution  following  be  used,  the  center  line  of  the  turnout 
curve  when  produced  back  until  parallel  to  the  main  track  will 
not  lie  on  the  center  line  of  the  main  track,  but  the  slight 
distance  from  the  center  may  be  calculated  and  an  allowance 
made,  so  that  the  details  of  turnouts  to  parallel  lines  can  be 
calculated  with  simplicity. 


86 


Railroad  Curves  and  Earthwork. 


154.  Problem.  Given,  in  a  turnout,  the  gauge  of  track  g, 
length  of  switch  rail  I,  the  distance  be- 
tween rails  t,  and  frog  angle  F. 
Required  the  radius  of  turnout  track  and 
distance  from  movable  end  of  rail  to  point 
of  frog  (or  the  "lead"). 

I        H 

Let  El  F  and  QK  be  the  rails  of 
the  turnout. 

Draw  M I  parallel  and 

OM  perpendicular  to  QF. 

Let  S  =  switch  angle 

t  =  HI 


^    cos  F 


•D   _j_  y    

~~ 


g-t 


2     cos  8  —  cos  F 

QF=     QD     +      DF 
ID 


(114) 


=       I     .+ 
E  =      I       + 


tanlFD 

g-t 


(115) 


Some  prefer  even  greater  precision,  and  give  weight  to  the 
fact  that  the  frog  is  straight,  not  curved.  It  is  customary 
apparently  to  make  the  switch  rail  straight  even  when  used  on 
curves,  and  it  is  not  customary  to  use,  for  curved  main  line, 
formulas  derived  on  as  strict  and  perfect  a  basis  as  is  shown  in 
(114),  (115)  and  in  (116)  and  (117)  shown  below.  It  does  not 
appear  that  the  requirements  of  the  work  demand  such  exces- 
sive refinement  of  calculation  as  would  be  necessary  to  do  this. 


Turnouts.  87 

155.  "Where  the  curve  continues  beyond  the  point  of  frog, 
the  introduction  of  a  short  tangent  (formed  by  the  frog)  hurts 
the  alignment  so  far  as  appearance  is  concerned,  and  also  makes 
less  convenient  the  office  work  of  laying  out  yards  and  other 
turnout  work.  As  a  matter  of  fact,  no  advantage  results  so  far 
as  the  easy  running  of  the  train  is  concerned.  The  path  traveled 
by  the  center  of  a  truck  is  here  fixed,  not  by  the  line  of  the  outer 
rail,  as  is  the  rule  on  curves,  but  by  the  position  of  the  guard 
rail  near  the  inside  rail,  and  the  actual  path  is  likely  to  be  a 
double-reversed  curve  of  some  sort,  being  fixed  finally,  in  large 
part,  by  the  gauge  of  the  inside  of  the  car  wheels,  and  thus 
varying  slightly  with  different  wheels.  If  it  be  considered 
essential  to  reconcile  mathematically  the  position  of  the  frog 
(as  to  length  of  lead)  with  the  form  of  the  frog,  the  best  way, 
probably,  is  to  make  the  frog  curved  (on  both  sides)  to  suit  the 
curvature  of  the  regular  turnout  curve.  The  result  will  be  that 
the  frog  will  truly  fit  the  turnout  curve,  while  on  the  straight 
line  the  frog  point  will  lie  outside  the  proper  gauge  line  by  a 
desirable  amount,  rendering  it  possible  to  make  the  path  of  the 
center  of  the  truck  nearly,  or  quite,  a  straight  line. 


156.  Problem.  Given,  in  a  turnout,  the  gauge  of  track  g, 
length  of  switch  rail  I,  distance  between 
rails  t,  length  of  frog,  end  to  point,  k, 
and  the  frog  angle  F. 

Required  the  radius  of  turnout  curve,  and 
the  distance  from  movable  end  of  rail  to 
point  of  frog. 

p           M  Then  following  the  methods  of  the 
preceding  problem, 


88 


Railroad  Curves  and  Earthwork. 


157.  Two  parallel  straight  tracks  may  be  conveniently  con- 
nected by  a  turnout  in  four  different  ways  : 

I.   By  a  reversed  curve,  the  two  curves  having  equal  radii. 
II.    By  a  reversed  curve,  the  two  curves  having  unequal  radii, 
and  with  P.  R.  C.  at  point  of  frog  F. 

III.  By  (a)  a  simple  curve  to  F, 

(&)   tangent,  and  return  by 

(c)    simple  curve  of  radius  equal  to  the  first. 

IV.  By  (a)  a  simple  curve  to  F, 

(b}  tangent,  and  return  by 

(c)   simple  curve  of  radius  unequal  to  first. 

158.  I.   Problem.     Given  the  perpendicular  distance  between 

two  parallel    tangents,  p;    also    the 
common  radius,  R. 

Required  Ir. 

i  « 
Formula  (71), 


159.   II.   Problem. 


Given  the  radius  of  the  first  curve,  RI, 

also  F  and  p. 
Required  the  radius  of  the  second  curve 

R%,  to  connect  the  parallel  tangents. 

If  P.  R.  C.  be  taken  at  F. 
Then  Ir   =  F 

UT       =  US  -  TS 
vers  TPF  =  US  -  TS 

vers   F   =  p  —  g 


160.   Problem.    Given,  as  above,  RI,  F,  p,  n. 

Required  R2. 
Second  Method. 


by  (96) 


(119) 


Turnouts. 


89 


161.   III.  Problem.     Given  R,  F,  p. 


I JT^ 


Required  the  length  I  of 
tangent  between  the  two 
curves  of  equal  radii. 

Let  AW,  ND  be  the  cen- 
ter lines  of  the  parallel 
tracks,  and  ABCD  the  turn- 
out. 

Draw  the  perpendicu- 
lars LB,  MCS,  BT. 


Then      BJ  =LM=W-          AL          -  MN 

CBsinBCT  =  AN-  AO  versAOF  -  PD  vers  DPC 
I   sin  F    =  p  -     fivers  F      -  R  vers  F 

I    —  P  —2R  vers  P 
siuF 


(120) 


In  the  case  of  a  cross-over  between  tracks,  it  will  be  conven- 
ient to  calculate  the  distance  from  F  to  H.  Both  frog  points 
can  then  be  located  and  the  entire  turnout  staked  out  without 
transit. 

162.   IV.  Problem.     Given  R^  g,  p,  I,  F. 

Required  R%. 

Let  EN  and  QL, 
and     TW  and  DG,  be  the 
rails  of  the   parallel  tan- 
gents, and  EFST  and  QCRD 
the  rails  of  the  turnout. 
T  w      Draw    the    perpendicu- 
lars US,  SM. 
D   G 
Then      SU  =  LM  =  NT  -  NL  -MT 

FS  sin  UFS  =  NT  -  NL  -  PS  versSPM 

I  sin  F  =  p   -  g  -  ( R*  -|J  vers  F 

w  _*    P  -  g  - 


90 


Railroad  Curves  and  Earthwork. 


From  (71) 


Similarly, 


163.  Problem.  Given 
for  tracks  as  shown  in 
figure,  the  radius  R  of 
turnout  curve,  also  the 
perpendicular  distances 
between  tracks  p,  p',  p". 

Required  BC,  CD. 


vers  AOB  = 

B 

BC  sin  CBE  =  CE 

BC  sin  AOB  =  p' 

BC  = 


(122) 


sin  AOB 


164.  Problem.  Given  the  radial  distance  p  between  a  given 
curved  main  track  and  a  parallel  siding, 
also  frog  angle  F  (or  number  n)  and  gauge 
of  track  g. 

Required  the  radius  of  second  curve  to  con- 
nect point  of  frog  with  siding. 

I.   When  the  siding  is  outside  the  main  track.  • 

Let  CM  be  the  inner  rail  of 
the  given  main  line. 

CFT  inner  rail  of  turnout. 

JROT  =  radius  of  main  line  (cen- 
ter). 

Rt  =  radius  of  turnout  (cen- 
ter). 

p  =  TN  =  radial  distance. 

Connect  FT,  FO. 
Let  FOT  =  0. 


Turnouts.  91 

In  triangle  FTO,  FO  =  Em  + 


also  OFT  +  OTF  =  =  180°  -  O 

OFT  -  OTF  =  OFT  -  PFT  =  F 
Then 

tan  KOFT  +  OTF)  :  tan  £(OFT  -  OTF)  =  TO   +  FO  :  TO  -  FO 
cot  i  0  :  tan  \F  =  2Rm  +  p     :  p   —   g 


p  —  g      co 


(123) 


Similarly  FPT  =  J?1  +  O 

Join  FS. 

In  the  triangle  PFS,  ~ 


(p  - 


Length  of  curve  L  =  100(F+  °)  (125) 

Dt 

165.   Approximate  Method. 

It  might  readily  be  shown  that  if  the  entire  turnout  be  cal- 
culated as  if  from  a  straight  track,  using  the  same  values  of 
n  and  p,  and  the  degree  of  each  curve  (Z>i,  D2)  be  found  ;  then 
it  would  be  approximately  true  that,  in  the  case  of  a  curved 
main  track,  the  degrees  of  the  turnout  curves  required  would 
be  found  by  adding  or  subtracting  Dm  to  or  from  D\  and  Z)2. 
The  distances  CF,  FT  would  also  be  the  same  as  in  the  turn- 
out from  straight  track.  The  demonstrations  would  follow  in 
principle  closely  those  given  in  reaching  (104),  (105). 


92 


Railroad  Curves  and  Earthwork. 


166.   Example. 

Turnout  from  curve  outside  the  main  track. 

Let'Dm  =  4  ;  n  =  0  ;  p  =  15  ;  g-  4.7. 

Precise  Method. 

(p  -  p)  n   10.3       92.7 
tan|0  =  ^  ^  =  ___x9  =  n^ 


•**'   2"tan'KJ?'+C 
92.7 


92.7 
1440.2 

£0  =  3°  40'  58" 
£  ^=3°  10'  47" 
0)  =  6°  51 '45" 
92.7 
770.3 

±p=   7.5 
U,  =  777.8 

Dt  =  7°  22'  17" 


L      100(1^  +  Q)  =  100  x  13°  43'  30"  = 
i>,  7°  22' 17" 


tan  6°  51  '45" 


log  1.967080 
log  3.158422 
tan  8. 808658 

tan  9.080444 
log  1.967080 
log  2.886636 


167.   Approximate  Method. 

In  the  case  of  a  turnout  from  a  straight  main  track,  where 
n  =  9  and  p  =  15, 


=  (15.0  -  4.7)2  x  81  =  1668.6 
H  =  1676.1  ;  D  =  3°  25'  ;  F  =  6°  22'  (Table  XXII.) 

L  =  10°  ^  6°  22'  =  186.3  for  straight  tracks 


=  3°  25'  +  4°  =  7°  25'  (7°  22'  precise  method) 
L  =  186.3  as  with  straight  track  (186.2  precise  method) 


Turnouts. 


93 


168.    II.  When  the  siding  is  inside 
P    the  main  track. 

In   a   similar   fashidn   it   may    be 
shown,  using  this  figure,  that 


From  triangle  OFT 

(P  - 


tan  |  O  = 


From  triangle  PFS 
Rt  -^  =  — 


-  0) 


(126) 


(127) 


(128) 


From  triangle  PFS 


169.  III.  When  the  siding  is  out- 
side the  main  track,  but  with  the 
center  of  turnout  curve  inside  of  main 
track. 

Let  EFS  be  the  outer  rail  of  main 
track. 

FT     the  inner  rail  of  turnout. 

From  triangle  OFT 

tan  i  0=^-^        (129) 
7?     4.  ^ 
Mm+  2 


p_     (p-g}n 
2 


100CP+Q) 

A 


(130) 


(131) 


With  both  §  168  and  §  169,  approximate  formulas  may  be 
used,  the  method  being  similar  to  that  of  §  165.  Experience 
will  determine  in  what  cases  it  will  be  sufficient  to  use  the  ap- 
proximate results,  and  where  precise  formulas  should  be  used. 


94 


Railroad   Curves  and  Earthwork. 


170.  Problem. 


Let 


Given  .the  radial  distance  between  a  given 
curved  main  track  and  a  parallel  siding. 
The  two  tracks  are  to  be  connected  by  a 

cross-over,  which  shall  be  a  reversed  curve 

of  given  unequal  radii. 

Required  the  central  angle  of  each  curve  of 
the  reversed  curve. 


ARB  be  center  line  of  turnout. 
AC  center  line  of  main  track. 


Then  in  the  triangle  POQ 
PO  =  Rm  +  7?i 
PQ  =  Si  +  R2 
OQ  =  OC  +  CB  -  BQ 

.      }  =  Rm  +    p     -  R2 

Solve  for  OPQ,  PQO,  POQ,  then 
RQB.  In  practice  this  problem  might 
take  the  following  form  :  Given  Rm, 
p,  g.  Assume  n  (or  F)  and  n1  (or  F'} .  From  these  calculate 
RI  and  R2  (or  use  Table XXII.,  Allen).  Then  solve  as  above. 

171.   Approximate  Method. 

Where  p  is  very  small  compared  with  Rm,  the  degree  of  curve 
used  will  frequently  be  found  by  the  formulas  (approx.) 

and  Dta  =  D  -f  Dm 

The  length  of  each  part  may  be  found  for  a  cross-over  between 
parallel  straight  tracks,  using  the  same  values  of  n  andp,  and  the 
same  lengths  used  for  this  cross-over  between  curves. 

The  process  is  similar  in  every  way  to  that  shown  by  example 
in  the  previous  problem. 

A  similar  method  of  treatment  will  be  applicable  in  all  turn- 
outs from  curves  where  the  distance  between  tracks  is  not  too 
great. 


CHAPTER  IX. 
"Y"   TRACKS  AND   CROSSINGS. 

172.  In  many  cases  where  a  branch  leaves  a  main  track, 
an  additional  track  is  laid  connecting  the  two.  This  is  called 
a  "Y"  track,  and  the  combination  of  tracks  is  called  a  UF." 


173.  Problem.  Given  a  straight  main  track,  also  the  P.  C. 
and  radius  of  a  simple  curve  turnout. 
Also  radius  of  "T"  track. 
Required  the  distance  from  P.  C.  of  turnout 
to  P.O.  of  "F"  track;  also  the  central 
angles  of  turnout  and  o/"]F"  track  to 
the  point  of  junction. 


Let  AC    be    the    given 
straight  main  track. 

AB  the  turnout. 
CB  the  "F"  track. 

Draw  perpendicular  NP. 
i 

AC  =  I 
AOB  =  It 
CPB  =  Iy  =  180°  -  It 


Let 


Then 


(132) 
(133) 


96 


Railroad  Curves  and  Earthwork. 


174.   Problem. 


Given  a  straight  main  track,  also  the  P.  C., 
radius,  and  central  angle,  of  simple  curve 
turnout  connecting  with  a  second  tangent  ; 
also  the  radius  of  "  T"  track. 
Required  the  distance  from  P.  G.  of  turnout 
to  P.O.  of  "  Y"  track,  and  from  P.  T.  of 
turnout  curve  to  P.  T.  of"Y"  track. 

r  Let  AC  be  the  given  main 

track;    ABD  the  turnout;    CD 
the  "F"  track. 

Let  AO  =  Rt  ;     CP  =  Ry 
AC=Z;        BD  =  w 
Use  similar  notation  for  Tt  ; 


Produce  BD  to  E. 

Then              AC  =          AE  +          EC 

=  AO  tan  $  AOB  +  CP  tan  £  CPD 

I  =  Rt  tan  £  It    +  Ry  cot  \    It 

1=           Tt  +           Ty                        (134) 

BD=          ED  EB 

m=          Ty  Tt                        (135) 

175.    Problem.     In  the  accompanying  sketch  where 
ABC  =  main  track. 
AD  =  turnout. 

Given          AB  =  I 


Required  the  points  D  and  C. 

Find  PO,     PQ,     QO,  also  OPA 
then      POQ,  OPQ,  PQO 
then      BOC,  APQ 
D  and  C  will  then  be  easily  determined. 


"F"  Tracks  and  Crossings. 


97 


176.  In  the  figure  where  ABC  is  the  main  track  and  DC 

is  the  turnout. 

/D  Given     OB  =  Em 

BOC=  O 

Eequired  the  points  A  and  D. 
Find  QH,    OH 
then      EP 
then      EPQ,  EQ 
then  EH  =  AB, 

and  PQO  =  EQP  +  OQH 

PQO  determines  position  of  D 

EPQ  determines  length         AD 

« 

177.  Problem.     Given  a  curve  crossing  a  tangent,  and  the 

angle  C  between  tangent  and  curve  j  also, 

-R>  ff,  d'> 

Eequired  frog  angles  at  A,  B,  F,  D. 

Draw  AO,  BO,  CO,  FO,  DO  ;  also,  MO  perpendicular  to  CM. 
Then        MO  =  E  cos  C  a' 


cos  A  — 


(136) 


E 


cosZ>  = 


cos  B  — 


MO+I 
*  +  g 


MO  + 


pi 


(137) 


(138) 


(139) 


98 


Railroad  Curves  and  Earthwork. 


178.  Problem.  Given  radii,  JKi,  J?2,  of  two  curves  crossing 
at  C;  also  angle  at  crossing  (7;  also  g 
and  g'. 

Required,  frog  angles  at  A,  B, 
D,  E;  also  lengths  AB,  BE, 
DE,  AD. 

Find  in  triangle  OCR,  the 
line  OP. 

Find  in  triangle  OPA,  angles 
APO,  AOP,  and  GAP  =  A. 

Find  in  triangle  OPB,  angles 
BPO,  BOP,  and  OBP  =  B. 

Then  APB  =  BPO  -  APO. 

The  frog  angles  at  D  and  E, 
and  the  lengths  AD,  DE,  EB, 
may  be  calculated  in  similar 
fashion. 


CHAPTER  X. 
SPIRAL  EASEMENT  CURVE. 

179.  Upon  tangent,  track  ought  properly  to  be  level  across  ; 
upon  circular  curves,  the  outer  rail  should  be  elevated  in  accord- 
ance with  the  formula 

av2 

e  = 


32.2  .R 

in  which  e  =  elevation  in  feet 

g  =  gauge  of  track 
v  =  velocity  in  feet  per  second 
E  =  radius  of  curve  in  feet 
In  passing  around  a  curve,  the  centrifugal  force  C  = 


32. 


It  is  desirable  for  railroad  trains  that  the  centrifugal  force 
should  be  neutralized  by  an  equal  and  opposite  force,  and  for 
this  purpose,  the  outer  rail  of  track  is  elevated  above  the  inner. 
Any  pair  of  wheels,  therefore,  rests  upon  an  incline,  and  the 
weight  W  resting  on  this  incline  may  be  resolved  into  two  com- 
ponents, one  perpendicular  to  the  incline,  the  other  parallel  to 
the  incline,  and  towards  the  center  of  the  curve. 

We 

The  component  P  parallel  to  the  incline  will  be  P  =  — 

It  will  be  a  very  close  approximation  to  assume  that  C  acts 
parallel  to  the  incline  (instead  of  horizontally).  The  centrifugal 
force  will  be  balanced  (approx.)  if  we  make 

P=CorEf= 


g       32.2  E. 

<UO) 


99 


100          Railroad  Curvet  and  Earthwork. 

In  passing  directly  from  tangent  to  circular  curve,  there  is 
a  point  (at  P.  C.)  where  two  requirements  conflict  ;  the  track 
cannot  be  level  across  and  at  the  same  time  have  the  outer 
rail  elevated.  It  has  been  the  custom  to  elevate  the  outer  rail 
on  the  tangent  for  perhaps  100  feet  back  from  the  P.C.  This 
is  unsatisfactory.  It  is  therefore  becoming  somewhat  common 
to  introduce  a  curve  of  varying  radius,  in  order  to  allow  the 
train  to  pass  gradually  from  the  tangent  to  the  circular  curve. 

180.  The  transition  wffl  be  most  satisfactorily  accomplished 
when  the  elevation  e  increases  uniformly  with  the  distance  Z 
from  the  P.S.  (point  of  spiral)  where  the  spiral  easement  curve 

leaves  the  tangent  ;  then  -  is  a  constant 

" 


Since       g,  w,  A  are  constants,  SI  —  C 

TLen  El  =  BJe  and  E  =  ^  (141) 


Rt  =  radius  of  circle 
DC  =  degree  of  circular  curve 
le  =  total  length  of  spiral 

Let  «  =  the  "  Spiral  Angle  "  or  total  inclina- 

tion of  curve  to  tangent  at  any  point. 

8t  =  spiral  angle  where  spiral  joins  circle. 


Then 

M 


Jdl 


Again  dr  =  <flsin*      and      <Zy  =  <ffcos« 


Cubic  Spiral  Est&mftot-  Curve*  101 

All  values  of  *  will  generally  be  smaH,  and  we  »na/  assurae 

sin  s  =  *  and  cos  *  =  1 

then  dx  =  sdl  dy  =  dl 


Integrating,    *  = 

which,  is  the  equation  of  the  "  Cubic  Parabola,*'  a  curve  fre- 
quently used  as  an  easement  curve, 

If,  however,  the  approximation  coss  =  1  be  not  used,  the 
resulting  curve  will  be  more  nearly  correct  than  is  the  Cubic 
Parabola.  In  this  case  sins  =  * 


Integrating,         t  x 

The  resulting  curve  we  may  call,  for  the  lack  of  a  better 
name,  the  "Cubic  Spiral"  Easement  Curve. 

181.  The  Cubic  Parabola  is  well  adapted  to  laying  out  curves 
by  "offsets  from  the  tangent."  Modem  railroad  practice  favors 
-deflection  angles"  as  the  method  of  work  wherever  practi- 
cable. In  the  case  of  an  easement  curve  the  longitudinal  meas- 
urements are  most  conveniently  made  as  chords  along  the  curve, 

so  that  z  =  — — -  represents  a  curve  more  convenient  for  use 
than  is  x  =  ,  JL.    as  well  as  more  nearly  correct.    Evidently 


the  properties  of  the  two  curves  will  be  very  similar. 

The  proper  easement  curve  is  seen  to  be  a  curve  of  constantly 
(or  at  least  frequently)  changing  radius.  Searles,  Holbrook, 
Talbot,  and  Crandall  each  have  excellent  curves  of  this  char- 
acter. No  important  difference  exists  between  them,  except  in 
the  method  and  convenience  of  laying  out  the  curve,  and  in  this, 
the  arrangement  of  tables  is  an  important  feature.  Methods 
of  use  for  the  Cubic  Spiral  will  be  developed  in  the  following 


102 


'Railroad 'Curves  and  Earthwork. 


18ar,$teeM.  'in  a  Cubic  Spiral,  I,  ?c,  Rc 
Required  s,  sc,  and  "total  deflection  angles"  i,  ie 

72  0^. 

(142)  BGN=s  = 


This  is  the  expression  for  the  central  angle  of  the  connect- 
ing circular  curve  whose  length  is  one  half  the  length  of  the 
spiral.  It  also  follows  that  if  the  circular  curve  be  produced 
back  from  C  to  K  where  it  becomes  parallel  to  AN,  its  length 
will  be  ^,  since  KOC  =  CFN. 

Again  for  any  point  B  on  the  spiral 

sin  BAN  =  shu  =  -  (approx.) 


But 

Whence 

Also 


-«*«. - 


i  :  ic  =  I2  :  ?c2 
Also  the  back  deflection  ABG  =  BGN  -  BAN 


(146) 
(146.4) 


Also 


(146  B) 


Cubic  Spiral  Easement  Curve.  103 

183.    Given  Z,  Zc,  Rc 
Required  y  and  yc. 

From  (30)  the  excess,  c  -  a  =  — 

2C 

Let  e  =  c  —  a 

Then  in  the  Cubic  Spiral  the  excess  at  any  point  on  the  spiral 


- 


184.    Given    J2C,   yc,   %c,   sc 

Required  AL  =  q  and  LK  =  p. 

In  the  figure  CN  =  xc  and  AN  =  yc 

AL  =  AN  -  OCsinCOK 

q  =  yc    —  Rc  sin  sc  (148) 

LK  =  CN  -OCversCOK 
p  =  xc    —  Rc  vers  sc  (148  A') 

Values  of  ic  from  (146),  i  from  (146^1),  xc  from  (144), 
yc  from  (147-4),  q  from  (148),  p  from  (148^4)  are  computed 
and  shown  in  Table  XXXIII.  of  Allen's  Tables,  as  are  also 
values  of  D  computed  from  Dc  on  the  basis  that  D  is  directly 
proportional  to  the  length  of  curve  (141,4). 


104          Railroad  Curves  and  Earthwork. 

185.  Field-work  of  laying  out  Cubic  Spiral,  using  Table 
XXXIII. 

(a)  Select  on  ground  (or  fix)  point  L  opposite  the  point  K 
where  the  circular  curve  will 
become  parallel  to  tangent. 

(6)  Select  from  Table,  length 
of  spiral  to  join  given  circular 
curve.  The  tables  will  give,  for 
the  length,  some  multiple  of 
30ft. 

(c)  Fix  point  A  at  P.S.  of  spiral  (AL  =  q),  or  set  P.S.  at 
proper  distance  T8  from  vertex  V. 

(d)  With  transit  at  P.S.  use  chords  of  30  ft.  and  deflection 
angles  from  Table,  to  set  points  on  spiral,  including  the  P.C.C. 
atC. 

(e)  With  transit  at  P.C.C.  at  C,  turn  vernier  to  0,  and  beyond 
0  to  measure  angle  FCA  =  2  ic. 

(/)  Sight  on  P.S. 

(gr)  Turn  vernier  to  O,  and  thus  bring  line  of  sight  on  tangent 
to  curves  at  P.C.C. 

(ft)  Lay  out  circular  curve  as  usual,  or 

(i)  If  preferred,  run  out  circular  curve  with  transit  at  K, 
taking  backsight  on  an  offset  from  tangent  =  p. 

186.  Example.     Eequired  Cubic  Spiral,  270  ft.  long,   to 
join  4°  curve. 

Table  XXXIII.  gives  as  "deflection  angles,"  01',  05',  12', 
21',  33',  48r,  1°  05',  1°  25',  1°  48'.  The  angle  between  chord 
and  final  tangent  =  ACF  is  twice  1°  48',  the  final  deflection 
angle  =  3°  36'. 

187.  It  may  occasionally  (although  not  frequently)  happen 
that  the  entire  spiral  cannot  be  laid  out  from  the  P.S.,  and  it 
will  be  necessary  to  determine  deflection   angles  when  the 
transit  is  at  some  intermediate  point  on  the  spiral.     It  will  be 
desirable  to  occupy  some  regular  chord  point. 


Cubic  Spiral  Easement  Curve. 


105 


In  any  Cubic  Spiral,  the  degree  of  curve  D  increases  uniformly 
with  the  length  (141  A).  Hence 
the  degree  of  curve  at  I  must  be 
equal  to  the  difference  in  degree 
between  the  circular  curve  and 
the  spiral  at  5  where  length  A  I 
=  C5. 


P.S. 

j— 

A 


The  offset  from  tangent  or  tangent  deflection  E  I  will  therefore 
be  the  same  as  the  offset  from  the  circular  curve  D  5,  for  the 

tangent  deflection  a  =  —  (26)  is  inversely  as  the  radius  E,  or 

directly  as  the  degree  D,  whence  the  difference  in  tangent 
deflections  between  two  curves  D\  and  D%  will  be  the  same  as 
the  tangent  deflection  for  a  curve  whose  degree  is  D\  —  D%. 

It  will  further  follow  if  E  I  and  D  5  are  equal,  and  at  equal 
distances  from  A  and  C  respectively,  that  the  angles  E  A  I  and 
DCS  will  be  equal  (closely).  In  other  words,  the  divergence  of 
any  spiral  for  a  given  distance,  is  the  same  either  in  offset  or 
in  angle,  whether  the  divergence  be  from  the  tangent  or  from 
the  circular  curve.  From  this  it  also  follows  that  LT  =  TK. 

> 

188.  It  will  therefore  follow  that  if  at 
any  point  B  on  the  spiral  ABC,  the  transit 
toe  set  up  and  the  line  of  sight  be  brought  on 


the  auxiliary  tangent  BG  at  that  point,  then  the  deflection  angle 
to  any  forward  point  on  the  spiral  will  be  the  sum  of  (1)  the 
"total  deflection  angle,"  for  the  distance  from  B  to  that  point, 
due  to  the  circular  curve  HBJ,  whose  degree  is  the  degree  of  the 
spiral  at  B;  and  (2)  the  "total  deflection  angle"  from  the 


106          Railroad  Curves  and  Earthwork. 

original  tangent  for  that  spiral  for  the  same  distance  reckoned 
from  the  P.  8.  For  any  back  point,  the  deflection  angle  from 
this  auxiliary  tangent  will  be  the  difference  between  these 
angles. 

The  proper  use  of  these  deflection  angles  will  allow  the  line 
of  sight  to  be  brought  on  the  auxiliary  tangent,  as  well  as  give 
means  for  setting  all  points  on  the  spiral.  Table  XXXIII.  gives 
values  of  D  at  chord  points  30  ft.  apart. 

Example.  Kequired  forward  deflection  angles  from  point  6 
on  a  spiral  270  feet  long,  to  join  4°  curve. 

Table  XXXIII.  gives  Z>  at  point  6  =  2°  40'. 

Deflection  angle  for  30  ft.  on  2°  40'  curve        =      24'. 
The  total  angles  will  be  at  point  7,    24'  -f  01'  =      25', 

8,  48'  +  05'  =      53', 

9,  72'  -t-  12'  =  1°  24'. 

The  tangent  BG  is  found  by  laying  off  from  chord  AB,  twice 
the  forward  deflection  to  point  6,  or  2  x  48'  =  1°  .36'. 

189.   Problem.     Given  Dc  and  p 
Required  to  lay  out  Cubic  Spiral. 


From  the  equation  of  the  Cubic  Spiral 


Find  the  chord  length  KC  when  CQ  =  Bp  for  the  curve  of 

given  degree  Z)c,  taking  a  =  —  -  or  using  Table  VI.     Take  AL 
2  H 

and  also  length  of  curve  KC  equal  to  chord  KC,  and  find  corre- 
sponding central  angle  =  sc.    The  deflection  angle 

CAN  =  4  = 


Cubic  Spiral  Easement  Curve.  107 

Other  deflection  angles  may  be  found  by  the  formula 


The  back  deflection    ACF  =  2  ic 

If  the  chord  lengths  be  taken  =  —  the  required  angles  may 

be  taken  directly  from  Table  VII.  The  actual  numerical  com- 
putation of  angles  proportional  to  the  squares  of  the  distances, 
while  simple  in  principle,  will  be  found  somewhat  burdensome 
in  practice.  The  use  of  Table  VII.  is  therefore  recommended. 

As  an  alternative,  I  may  be  assumed  and  CQ  calculated,  and 
hence  p. 

Since  either  the  offset  p  or  the  length  I  may  be  selected  of 
any  desired  length,  this  method  makes  the  Cubic  Spiral  abso- 
lutely flexible. 

Where  great  precision  is  required  q  =  -  is  not  quite  exact, 

and  a  correction  may  be  made  of  the  same  amount  as  shown 
in  Table  XXXIII.  for  a  spiral  of  similar  dimensions.  A  correc- 
tion may  also  be  made  for  the  difference  between  the  length  of 
curve  KC  and  of  chord  KC  if  desired. 

190.   Example. 

Degree  =  4°.    Offset  =   1.90 

_  3          r  Since  4°  is  ^ 

Offset  from  tangent  4°  =   5.70(0.4  J  of  10°,  divide 

Offset  from  tangent  10°  =  14.25          I  by  0.4 

Table  VI.     Chord  length         =  128.0 

2 

Length  of  spiral  =  256.0 

Central  angle  128  ft.  4°  curve  =  5.12° 

=  5°7.2'  =  307.2' 
Deflection  angle  =  £  x  307.2   =  102' 

Table  VII.  gives  deflection  angles  for  chord  lengths  of 
TV  X  256  =  25.6  ft.  as  follows  :  1'  ;  4'  ;  9'  ;  16.5'  ;  25.5'  ;  36.5'  ; 
50'  ;  65.5'  ;  82.5'. 


108          Railroad  Curves  and  Earthwork. 

191.     Given  Dc  and  p, 

Required  to  lay  out  Cubic  Spiral  by  offsets  from  the  tangent. 

Proceed  as  before  and  find  xc  -  4p  or  use  Table  XXXIII. 

Find  also  I  as  in  §  186. 

Find  other  values  of  x  at  convenient  intervals  by  formula 


*=*•(£)' 


In  general  the  spiral  will  be  laid  out  by  deflection  angles  in 
preference  to  offsets. 

192.  Compound  Curves.  In  the  case  of  Compound  Curves, 
it  is  proper  and  desirable  that  easement  curves  should  be  intro- 
duced between  the  two  circular  curves  forming  the  compound 
curve. 


Problem.     Given  in  a  Compound  Citrve,  DI,  Z>s,  p,  or  I. 

Required  the  Deflection  Angles  for  a  Cubic  Spiral 
to  connect  the  circular  curves. 

(a)  Find  by  §  188  or  §  190  the  Deflection  Angles  proper  for 
a  Cubic  Spiral  to  connect  a  tangent  with  a  circular  curve  of 
degree  =  Z>j  —  Ds. 

Let  these  =  u,  t'2,  i%,  etc. 

(6)  Find  the  deflection  angles  to  corresponding  points  on 
one  of  the  circular  curves,  the  auxiliary  tangent  for  these  being 
at  the  point  where  the  Cubic  Spiral  leaves  this  circular  curve 
(where  the  transit  will  be  set). 

Let  these  =     ,      ,     ,  etc. 


Cubic  Spiral  Easement  Curve.  109 

(c)  The  total  deflections  required  will  be  for 
point  1  —  +  ii 

point  2  7^  +  1*2 

point  3  ff  +  ^  etc- 

Similar  procedure  may  be  followed  if  it  be  desired  to  lay  out 
the  spiral  by  offsets.  Convenient  points  may  be  set  on  the 
circular  curves  and  the  offsets  taken  from  either  curve. 

193.    Given  7,  I,  and  7fc  or  Dc. 

Required  the  Tan- 
gent Distance  Ts. 

From  Table  XXXIII. 
(or  by  §  184)  find  q  and  p. 

(a)  When  the  same  spiral  is  used 
at  both  ends  of  the  circular  curve. 


AV  =  AL+          LU      +UV 

=  AL+         Kl      -fUltanUIV 
=    q  -f  Ec  tan  \  I  +  p  tan  |  7 
Tt  =    q  +  Tc  +  p  tan  £  7  (149) 

where  Te  is  tangent  distance  for  circular  curve  alone. 

,H u_V \b~)  When  different  spirals  are  used  at 

the  ends  of  the  circular  curve. 

In  this  case  a  new  value  must  be  found 
for  UV  and  also  for  UW. 

Let       Ul  =pi  and  IW=p2 
FIG.  A. 

VU=      HV     -       HU 
IW  Ul 


sinVHI      tanUHl 

P2  Pi 

sin  /          tan  I 


110 


Railroad  Curves  and  Earthivork. 


FIG.  A. 


H       V       U 

-,- ^ r 

!\\! 

i    V>" 

"ML          iv"* 


FIG.  B. 


Let 

In  Fig.  A 


In  Fig.  B 


Also 


[}\=pi  and   IW=p2  and   HM  =pi 
VW=     VG     -      WG 
_      Ul  IW 

"sinVGI      tanWGI 

Pi  Pz 

sin  /       tan  / 

i     T»      i       ^1  -P^  /"MQ   Z?\ 

^2  +  ^c  +  -;-^--r— ->  (1495) 


VU  =     HU      -  HV 

_      Pi  Pz 

tan  /       sin  / 

T1      —        ,         i       T>  P^- 


Sl        ic  -tan/ 
VW  =  VG  -  WG 


sin  /     tan  / 


sin  /     tan  / 


(149  J5) 


194.   Problem. 

q  and  p  of 
spiral. 

Required 

the  distance  BH  =  h 
through  which  the  cir- 
cular curve  GHE  must 
be  moved  in  along  VO 
to  allow  the  use  of  this 
spiral ;  also  this  dis- 
tance GA  =  d  from  P.O. 
to  P.S. 


I  and  Bc  for  circular  curve  GHE, 

L       G 


X* 


Cubic  Spiral  Easement  Curve. 
KL 


BH  =  PO  =  KG  = 


cos  LKG 
P 


111 


(150) 


GA  =  AL  +  LG 

=  AL  +  LK  tan  LKG 
d  =    q  4-  p  tan  £  / 


(150 .4) 


195.  Problem.  Given  land  EI  for  circular  curve  DB  ;  also  q 
and  p  of  spiral ;  also  BH 
=  h  measured  along  VO 
locating  H  through  which 
new  circular  curve  is  to 
pass  to  allow  the  use  of 
this  spiral. 

Eequired  the  radius 
EZ  =  KP  of  the  new  curve 
KH  ;  also  distance  Dk  =  d 
from  P.O.  to  P.S. 

PO  =  NO  =  OB  +  BH  -PH 

=  EI  +  h  -EZ 

OM'=  DO  -  DM 

=  DO-PK  -  KL 

NM  =  NO  -  OM  =  h  4 
PO  versNOP  =  NM 
(J?i  —  EZ  +  K)  vers  \I     =h  +p 

T>  73      ,     ;,  _    h  4-  p 


vers  £  / 

DA  =  AL  -  DL 
.    =AL-MP 


(151) 


112 


Railroad  Curves  and  EartTiivork. 


196.   Problem.     Given  I  and  R\  of  circular  curve,  also  q,  p, 

and  sc  of  spiral,  also 
angle  CPF  =  72, 

-6 2^Jr-- —    through  which   the 

circular  curve  is  to 
pass  when  changed 
and  made  sharper 
to  allow  the  use  of 
the  given  spiral. 

Required  the  ra- 
dius Hz  of  new  curve 
CF,  to  compound 
with  original  curve 
FH  ;  also  the  dis- 
tance DA  =  d  from 
P.O.  to  P.S. 

OP  vers  NOP  =  NM  =  MD  -  ND 

=  LP  -KP=p 


Si  -  R*  = 


Also 


vers  (72  + 
DA  =  AL-DL 
=  AL-MP 
d  =  q  -  (Hi  - 


(152) 


sin  (J2  +  sc)  (152  A) 


CHAPTER  XI. 
SETTING  STAKES  FOR  EARTHWORK. 

197.  The  first  step  in  connection  with  Earthwork  is  staking 
out,  or  "  Setting  Slope  Stakes,"  as  it  is  commonly  called. 

There  are  two  important  parts  of  the  work  of  setting  slope 
stakes : 

I.   Setting  the  stakes. 

II.  Keeping  the  notes. 

The  data  for  setting  the  stakes  are  : 

(a)  The  ground  with  center  stakes  set  at  every  station  (some- 
times oftener). 

(&)  A  record  of  bench  marks,  and  of  elevations  and  rates  of 
grades  established. 

(c)  The  base  and  side,  slopes  of  the  cross-section  for  each 
class  of  material. 

In  practice,  notes  of  alignment,  a  full  profile,  and  various 
convenient  data  are  commonly  given  in  addition  to  the  above. 

198.  I.  Setting  the  Stakes.     The  work  consists  of : 

(a)  Marking  upon  the  back  of  the  center  stakes  the  "cut" 
or  "  fill "  in  feet  and  tenths,  as 

C  2.3  or  F  4.7. 

(&)  Setting  side  stakes  or  slope  stakes  at  each  side  of  the 
center  line  at  the  point  where  the  side  slope  intersects  the  sur- 
face of  the  ground,  and  marking  upon  the  inner  side  of  the 
stake  the  "  cut "  or  "  fill "  at  that  point. 

113 


114         Railroad  Curves  and  Earthwork. 

199.    (a)  The  process  of  finding  the  cut  or  fill  at  the  center 
stake  is  as  follows  : 

Given  for  any  station  the  height  of  instrument  =  hi,  and  the 
elevation  of  grade  =  hg. 

Then  the  required  rod  reading  for  grade 

rg  =  hi-  hg.  (153) 

It  is  not  necessary  to  figure  hg  for  each  station. 

Let  hg0  =  hg  at  Sta.  0 

hffi  =  hg  "    "     I 
hgt  =  hg  "    "    2,  etc. 

Also  use  similar  notation  for  rg. 

Let  g  =  rate  of  grade  (rise  per  station) 

Then  hffl  =  hffo  -f  g 

kgt  =  V  +  0 

hg3  =  hff2  +  gr,  etc. 


=  hi  -  h 


rffl  =  rgo-g  (154) 

Similarly,      r<,2  =  rffi  —  g,  etc. 

It  will  be  necessary,  or  certainly  desirable,  to  figure  hg  and 
rg  anew  for  each  new  ht.  It  is  well  to  figure  hg  and  rg  Tas  a 
check)  for  the  last  station  before  each  turning  point. 


Setting  Stakes  for  Earthwork.  115 

200.   Example.  ft,  =  106.25 

Sta.  0,  grade  elevation  100.00  Q() 


10,      "  "         107.50 

r,0  =  106.25  -  100.00  =  6.25  6.25 

g  \ 
rffi=  6.25  -  1.00  j  =5.25 

r9z  =  5.25  -  1.00  =  4.25 

r9z  =  4.25  -  1.00  =  3.25 

rffi  =  3.25  -  1.00  =  2.25 

rg_  -  2.25  -  1.00  =  1.25 

Change  in  rate 

rg6=  1.25-0.50  =0.75 

»V7=  0.75-0.50     =0.25 

It  is  found  necessary  to  take  a  T.P.  here,  and  we  therefore 
find  hff7  =     hg5    +     2g 

=  105.00  +      1.00  =  106.00 
rffi  =  hi  -  hgj  =  106.25  -  106.00  =     0.25 

Therefore  all  intermediate  values  rg^  rg^  etc.,  are  "  checked." 

201.    Having  thus  found  rg,  next,  by  holding  the  rod  upon 
the  surface  of  the  ground  at  the  center  stake,  the  rod  reading 

M      '  rc  =  LO  is  observed  from 
~hi  the  instrument.     The  cut 
or  fill 

c  =  OG  =  MN-  LO 

=  rg  -  rc    (155) 

*—hg       In  the  figure  given  the 
values  of  rg  and  c  are  posi- 
tive ;  a  positive  value  of  c  indicates  a  "cut,"  a  negative  value 
of  I-  indicates  a  "fill." 
It  can  be  shown  that  in  the  two  cases  of  "  fill," 

(1)  When  hi  is  greater  than  hg,  and 

(2)  When  hi  is  less  than  hg, 

the  formula  given  will  hold  good  by  paying  due  attention  to  the 
sign  of  r9,  whether  +  or  —  . 


116         Railroad  Curves  and  Earthwork, 

202.   (6)  Setting  the  Stake  for  the  Side  Slope. 
(1)   When  the  surface  is  level. 


Let 


Then 


b  —  AB  =  base  of  section 
c  =  OG  =  center  height 


=OD  =  OE  =  distance  out 


=  £  b  +  sc 

203.  Setting  the  Stake  for  the  Side  Slope. 

(2)   When  the  surface  is  hot  level. 
Here  the  process  is  less  simple. 


(156) 


Let  b  =  AB  =  base 

c  =  OG  =  center  height  (or  cut) 
s  =  slope 


Setting  Stakes  for  Earthivork.  117 


hr  =  EK  =  side  height  right 
hi  =  DH  =  "        "      left 
dr  =  G  K  =  distance  out  right 
di  =  GH=       "         "    left 
Then  dr  =  4  6  4-  shr 


(157) 

=  \*  + 

But  hr  and  /^  are  not  known.     It  is  evident  from  the  figure 
that  hr  >  c  and  hi  <  c  in  the  case  indicated,  and  therefore 


204.  It  would  be  possible  in  many  cases  to  take  measure- 
ments such  that  the  rate  of  slope  of  the  lines  OE  and  OD  would 
be  known,  and  the  positions  of  E  and  D  determined  by  calcula- 
tion from  such  data.  But  speed  and  results  finally  correct  are 
the  essentials  in  this  work,  and  these  are  best  secured  by  find- 
ing hi  and  hr  and  the  corresponding  di  and  dr  upon  the  ground 
by  a  series  of  approximaXions,  as  described  below. 

Having  determined  c,  use  this  as  a  basis,  and  make  an  estimate 
at  once  as  to  the  probable  value  of  hr  at  the  point  where  the 
side  slope  will  intersect  the  surface,  and  calculate  dr  =  \  b  +  shr 
to  correspond. 

Measure  out  this  distance,  set  the  rod  at  the  point  thus  found, 
take  the  rod  reading  on  the  surface,  and  if  the  cut  or  fill  thus 
found  from  the  rod  reading  yields  a  value  of  dr  equal  to  that 
actually  measured  out,  the  point  is  correct.  Otherwise  make 
a  new  and  close  approximation  from  the  better  data  just  ob- 
tained, always  starting  with  hr  and  calculating  dr,  and  repeat 
the  process  until  a  point  is  reached  where  the  cut  or  fill  found 
from  the  rod  reading  yields  a  distance  out  equal  to  that  taken 
on  the  ground.  Then  set  the  stake,  and  mark  the  cut  or  fill 
corresponding  to  hr  upon  the  inner  side,  as  previously  stated. 

Perform  the  same  operation  in  a  similar  way  to  determine 
di  =  \  b  +  shi,  and  mark  this  stake  also  upon  the  inner  side  with 
a  cut  or  fill  equal  to  hi. 


118         Railroad   Curves  and  Earthwork. 


205.  It  requires  a  certain  amount  of  work  in  the  field  to  ap- 
preciate the  process  here  outlined,  but  which  in  practice  is  very 
simple.     It  may  impress  some  as  being  unscientific,  and  at  first 
trial  as  slow,  but  with  a  little  practice  it  is  surprising  how 
rapidly,  almost  by  instinct,  the  proper  point  is  reached,  often 
within  the  required  limits  of  precision  at  the  first  trial,  while 
more  than  two  trials  will  seldom  be  necessary,  except  in  difficult 
country. 

206.  The  instrumental  work  is  the  same  in  principle  as  at 
the  center  stake. 


Let 
then 


rr  =  NE  =  rod  reading  at  slope  stake  right, 
KN  -  NE  =  rg  -  rr  =  hr 


here  rg  is  the  same  for  center,  right  and  left  of  section. 

In  some  cases  it  may  be  necessary  to  make  one  or  more 
resettings  of  the  level  in  order  to  reach  the  side  stakes  from 
the  center  stake.  In  this  case,  of  course,  a  new  rg  must  be 
calculated  from  the  new  hi.  This  introduces  no  new  principle, 
but  makes  the  work  slower. 

207.  A  "slope-board"  or  "level-board"  may  be  used  to 
advantage  in  many  cases.  In  certain  sections  of  country  this 
might  be  considered  almost  indispensable.  It  consists  simply 
of  a  long,  straight-edge  of  wood  (perhaps  15  ft.  long)  with  a  level 
mounted  in  the  upper  side.  It  is  used  with  any  self-reading 
rod.  A  rod  quickly  hand  marked  will  serve  the  purpose  well. 
Having  given  the  cut  or  fill  at  the  center,  or  at  any  point  in  the 
section,  the  leveling  for  the  side  stakes,  and  for  any  additional 
points,  can  readily,  and  with  sufficient  accuracy,  be  done  by 
this  "level-board,"  and  the  necessity  for  taking  new  turning 
points  and  resetting  the  level  avoided. 


Setting  Stakes  for  Earthwork.  119 

208.  II.  Keeping  the  Notes. 

The  form  of  note-book  used  for  keeping  the  notes  of  slope 
stakes  and  of  center  cuts  and  fills,  often  called  "  cross-section" 
notes,  is  shown  on  the  following  two  pages. 

The  left-hand  column  for  stations  should  read  from  bottom 
to  top. 

The  surface  elevations  in  column  2  are  not  obtained  directly 
from  the  levels,  but  result  from  adding  to  the  grade  eleva- 
tion at  any  station  the  cut  or  fill  at  that  station,  paying  due 
attention  to  the  signs.  This  column  of  surface  elevations  need 
not  be  entered  up  in  the  field,  but  may  be  filled  in  as  office 
work  more  economically. 

The  column  of  grade  elevations  consists  of  the  grade  eleva- 
tions as  figured  for  each  station. 

The  figures  marked  +  are  cuts  in  feet  and  tenths,  and  those 
marked  —  are  fills  ;  the  figures  above  the  cuts  and  fills  are  the 
distances  out  from  the  center,  and  the  position  in  the  notes, 
whether  right  or  left  of  the  center,  corresponds  to  that  on  the 
ground. 

The  columns  on  the  right-hand  page  are  used  for  entering, 
when  computed,  the  "quantities,"  or  number  of  cubic  yards,  in 
each  section  of  earthwork. 

209.  The  column   "  General  Notes "   is  used  for  entering 
extra  measurements    (of   ditches,    etc.)    not    included  in  the 
regular  cross-section  notes;  also  notes  of  material  "hauled"  ; 
classification  of  material  and  various  other  matters  naturally 
classed  under  the  head  of  "Hem arks." 

210.  When  the  surface  is  irregular  between  the  center  and 
side  stakes,  additional  rod  readings  and  distances  out  are  taken, 
and  the  results  entered  as  shown  for  station  0  on  p.  118,  the 
section  itself  being  as  shown  below  in  the  sketch. 


Station  0 


120         Railroad  Curves  and  Earthwork. 
211.   Form  of  Crosf-Section  Book  (left-hand  page). 


(Date) 
(Names  of  Party) 
Base  20  ;  1  to  1 
14  ;  li  to  1 

Station 

Surface 
Elev. 

Grade 
Elev. 

Cross-Section 

5 

97.1 

105.00 

18.4                _79                              10.4 
-7.6                                                  ^8.3 

+69.7  P.  T. 

94.4 

104.70 

22,1                  1Q  g                             23.0 

-10.1                                                 -10.7 

4 

96.9 

104.00 

19.3               _71                            17.0 
—  8.2                                                —6.7 

+27.2  P.  C. 

98.0 

103.27 

-6.4                                                   -8.6 

3 

98.1 

103.00 

16.0               _49                             10.9 
-6.0                                                   -r-2.6 

+91 

100.6 

102.91 

13.3               _23                            10.0 
-4.2                                                       0.0 

+76 

102.8 

102.76 

10.3                   00                             11.9 
-  2.2                                                   +  1.9 

+64 

103.7 

102.64 

KM)               +11                            13.2 
0.0                                                  +3.2 

+50 

106.4 

102.50 

13.4                                                  17.1 

11                                               T  O.if                                                   

2 

115.1 

102.00 

J6.7             +131                        _26.7 
+  6.7                                               +  16.7 

1 

117.7 

101.00 

22.7     10.0  +  16  7                10.0      22.2 
+  12.7+17.2                         +13.1  +12.2 

0 

109.2 

100.00 

18.0       9.0     ,  9  o      8.5      18.4      24.6 
+  8.0+10.1          '     +7.8+14.7+14.6 

Setting  Stakes  for  Earthwork.  121 

212.    (Right-hand  Page.) 


Excavation 


L.  Rock       S.  Rock 


Earth 


Embank- 
ment 


General  Notes 


122 


Railroad  Curves  and  Earthwork. 


213.  Cross-sections  are  taken  at  every  full  station,  at  every 
P.  (7.  or  P.  T.  of  curve,  wherever  grade  cuts  the  surface,  and  in 
addition,  at  every  break  in  the  surface.  In  the  figure  below, 
showing  a  profile,  sections  should  be  taken  at  the  following 
stations  :  — 


19°  00 


At  Stations  0,  I,  2,  2  +  52,  3,  4, 
7,  8,  9,  9+29,  9  +  82,  10,  II,  11+30, 
13,  14,  15,  16,  17  P. T.,  18. 


5, 

12, 


5  +  80,      6, 
I2  +  25P.C'., 


214.  It  is  not  necessary  actually  to  drive  stakes  in  all  cases 
where  a  cross-section  is  taken  and  recorded,  but  in  every  case 
where  they  will  aid  materially  in  construction  stakes  should  be 
set.  It  is  best  to  err  on  the  safe  side,  which  is  the  liberal  side. 
In  passing  from  cut  to  fill,  it  is  customary  to  take  full  cross- 
sections,  not  only  at  the  point  where  the  grade  line  cuts  the 
surface  at  the  center  line  of  survey,  but  also  where  the  grade 
cuts  the  surface  at  the  outside  of  the  base,  both  right  and  left, 
as  in  the  figure  below,  which  illustrates  the  notes  on  p.  118 ; 
full  cross-sections  are  taken  not  only  at  stations  2  +  76,  but 
also  at  2  +  64  and  2  +  91. 


Setting  Stakes  for  Earthwork. 


123 


215.  Stakes  are  actually  set  at  the  center  G  and  at  the 
points  A  and  B,  where  the  outside  line  of  the  base  of  Excavation 
cuts  the  surface.     It  is  not  customary  to  set  stakes  or  record 
the  notes  for  the  points  A'  and  B',  where  the  outside  line  of  the 
base  of  Embankment  cuts  the  surface.    The  stakes  at  A,  G,  and 
B  are  a  sufficient  guide  for  construction,  and  the  solidities  or 
"  quantities  "  would  in  general  be  affected  only  slightly  by  the 
additional  notes  if  they  were  made.    When  the  line  AGB  crosses 
the  center  line  nearly  at  right  angles,  it  would  not  be  necessary 
to  take  more  than  one  section  so  far  as  the  notes  are  concerned. 
It  is  well,  however,  to  set  the  stakes  A  and  B  exactly  in  their 
proper  position. 

216.  Wherever  an  opening  is  to  be  left  in  an  Embankment 
for  a  bridge  or  for  any  other  structure,  stakes  should  be  set  as 
in  the  figure  below  :  — 


At  A  and  B  (at  the  side  of  the  base  and  top  of  the  slopes  AF 
and  BH)  stakes  should  be  set  marked  "  Bank  to  Grade  "  ;  and 
at  F  and  H  (at  the  foot  of  the  slopes)  stakes  should  be  set 
marked  "  Toe  of  Slope."  Where  the  bank  is  high,  an  addi- 
tional stake  K  at  foot  of  slope  may  be  set  as  an  aid  to  construc- 
tion. The  stakes  at  D  and  E  should  also  be  set  as  ordinary 
slope  stakes. 

217.  The  "level  notes"  proper,  or  the  record  of  heights  of 
instrument,  bench  marks,  turning  points,  etc.,  used  in  setting 
slope  stakes,  are  usually  kept  separate  from  the  cross-section 
notes.  One  reason  for  this  is  that  level  notes  run  from  top  to 
bottom  of  page,  while  cross-section  notes  read  from  bottom  to 
top  of  page.  The  level  notes  should  be  kept  either  in  the  back 


124         Railroad  Curves  and  Earthwork. 

of  the  cross-section  book  or  in  a  level  book  carried  for  that 
purpose.  Keeping  these  or  any  other  notes  on  a  slip  of  paper 
is  bad  practice. 

218.  Earthwork  can  be  most  readily  computed  when  the 
section  is  a  "Level  Section,"  that  is  when  the  surface  is  level 
across  the  section ;  but  this  is  seldom  the  case,  and  for  purposes 
of  final  computation  it  is  not  often  attempted  to  take  measure- 
ments upon  that  basis. 

219.  In  general,  in  railroad  work,  the  ground  is  sufficiently 
regular  to  allow  of  "  Three-Level  Sections"  being  taken,  one 
level  (elevation)  at  the  center  and  one  at  each  slope  stake,  as 
shown  by  these  notes,  where  Base  is  20,  and  Slope  £  to  1 :  — 

11.3  +42  12.8 


+  2.6  +5.5 

The  term  "  Three-Level  Section"  is  usually  applied  only  to 
regular  sections  where  the  widths  of  base  in  each  side  of  the 
center  are  the  same.  In  regular  three-level  sections  the  calcu- 
lation of  quantities  can  be  made  quite  simple.  To  facilitate 
the  final  estimation  of  quantities,  it  is  best  to  use  three-level 
sections  as  far  as  possible. 

220.  In  many  cases  where  three-level  sections  are  not 
sufficient,  it  may  be  possible  to  use  "Five-Level  Sections," 
consisting  of  a  level  at  the  center,  one  at  each  side  where  the 
base  meets  the  side  slope,  and  one  at  each  side  slope  stake,  as 
shown  by  the  following  notes  :  — 

Base  20,  Slope  1  to  1, 

22.7  10.0  7  10-°  22-2 


+  12.7          +  17.2  +  13.1          +  12.2 

The  term  "Five-Level  Section"  is  usually  applied  only  to 
regular  sections  where  the  base  and  the  side  slopes  are  the 
same  on  each  side  of  the  center. 

221.  Where  the  ground  is  very  rough,  levels  have  to  be 
taken  wherever  the  ground  requires,  and  the  calculations  must 
be  made  to  suit  the  requirements  of  each  special  case,  although 
certain  systematic  methods  are  generally  applicable.  Such 
sections  are  called  "Irregular  Sections." 


CHAPTER  XII. 
METHODS  OF  COMPUTING  EARTHWORK. 

222.  In  calculating  the  solidities  or  "quantities"  of  Earth- 
work, the  principal  methods  used  are  as  follows  :  — 

I.  AVERAGING  END  AREAS. 

II.  PRISMOIDAL  FORMULA. 

III.  MIDDLE  AREAS. 

IV.  EQUIVALENT  LEVEL  SECTIONS. 
V.  MEAN  PROPORTIONALS. 

VI.    HENCK'S  METHOD. 

223.  I.  Averaging  End  Areas. 
This  is  the  simplest  method :  — 


Station  1 


Station  0 


Let     AQ  =  area  of  cross-section  at  Station  0 

II  U  li, 


I  =  length  of  section,  Sta.  0  to  Sta.  I 

S  =  solidity  of  section  of  earthwork  (Sta.  0  to  I) 
125 


126         Railroad  Curves  and  Earthwork. 

Then  8  =  A°  ~^  Al  I  (in  cubic  feet)  (158) 

2 


_  AQ  + 


—  (in  cubic  yards) 


2          27 
As  (158)  is  capable  of  expression 


it  is  practically  based  on  the  assumption  that  the  solidity  consists 
of  two  prisms,  one  of  base  AQ  and  one  of  base  AI,  and  each  of 
a  length,  or  altitude  of  -• 


224.  To  use  this  method,  we  must  find  the  area  A  of  each 
cross-section  ;  the  cross-section  may  be  :  — 

(a)  Level. 

(6)  Three-Level. 

(c)  Five-Level. 

(c?)  Irregular. 

225.  (a)  Level  Cross-Section. 

Let   6  =  base          =  AB 
D  L  o  M          E 


c  =  center  ht.=  OG 
A  =  area  of  cross-section 


Methods  of  Computing  Earthwork.         127 


Then  DL  =  EM  =  sc 

A  =  AB'x  OG  +  DL  x  AL 
=  be  +sc2 

=  c(6  +  sc) 

226.    (6)  Three-Level  Section.    First  Method. 

M 


Let 


(160) 


<r^^^ 

o 

^•^S 

-^/ 

H                   A                  ( 

B                                                 Y 

Then^l= 


OGD 
x 


6  =  base  =  AB 

s  =  side  slope 

c  =  center  ht. 
hr  =  side  height  EK 
ht=    "        "      DH 
<?r  =  distance  out  M  E 
di=       "         "    DL 
A  =  area  of  cross-section 

+       OGE       +       QBE       +      AGO 

x  ME  +  |GB  x  EK  +  |AG  x  DH 


(161) 


128         Railroad  Curves  and  Earthwork. 

227.    (6)  Three-Level  Section.     Second  Method. 
M 


Using  the  same  notation. 


GV 


s       2s 


2s 


The  triangle  ABV  is  often  called  the  "  Grade  Triangle." 
Area  ABV  =  GV  x  GB 


4s 


Area  EODV  =  0V  x  —  +  0V  x 


A  =  EODV  -  ABV 
=  (c+-r- 


/ 


4s 


Let 


D  =  di  +  dr 


(162) 


In  using  this  formula  for  a  series  of  cross-sections  of  the 

same  base  and  slope,  -^-  and  —  are  constants,  and  the  compu- 
2s          4s 

tation  of  A  becomes  simple  and  more  rapid  than  the  first  method. 


Methods  of  Computing  Earthwork.         129 


228.    (c)  Five-Level  Section. 


Use  notation  the  same  as  before  ;  in  addition  let 
fr  —  height  MB 

/« =      "      LA 
Then  A  =  AOB  -f  DLOA  +  EMOB 

_     Cb       ,      frdr       ,    fldi 

'    2     +      2      +    2 


(163) 


229.    (d)  Irregular  Section. 

The  "Irregular  Section,"  as  shown  in  the  figure,  may  be 
divided  into  trapezoids  by  vertical  lines,  as  in  Fig.  1 ;  or  into 
triangles  by  vertical  and  diagonal  lines,  as  in  Fig.  2. 


y 

-r 

3   !  * 

i   T   i 

5  I'M 

•           i          \j 

FIG.  1. 


FIG.  2. 


The  triangles  in  Fig.  2  can  be  computed  in  groups  of  two 
each  pair  having  a  common  base  (vertical). 

It  will  be  seen  that  Fig.  1  requires  8  solutions  and  Fig.  2  only 
7  solutions  of  trapezoids  or  triangles.  The  computations  can 


130         Railroad  Curves  and  Earthwork. 

be  made  with  substantially  equal  simplicity  in  either  case,  and 
after  a  little  experience,  directly  from  the  notes  without  any 
necessity  for  a  sketch. 

230.  Another  method  which  has  been  used  for  calculating 
irregular  cross-sections  is  to  plat  them  on  cross-section  paper, 
and  get  the  area  by  "  Planimeter. "     In  very  irregular  cross- 
sections  this  method  would  prove  economical  as  compared  with 
direct  computation  by  ordinary  methods,  but  it  is  probable  that 
in  almost  every  case  equal  speed  and  equal  precision  can  be 
obtained  by  the  use  of  suitable  tables  or  diagrams  (to  be  ex- 
plained later)  ;  for  this  reason  the  use  of  the  planimeter  is  not 
recommended. 

231.  Having  found  the  values  of  A  for  each  cross-section, 
S  is  found  in  each  case  by  the  formula  above  given, 

^A±A.±(incUiyds>)  (159) 

It  is  found  that  this  formula  is  only  approximately  correct. 
Its  simplicity  and  substantial  accuracy  in  the  majority  of  cases 
render  it  so  valuable  that  it  has  become  the  formula  in  most 
common  use.  It  gives  results,  in  general,  larger  than  the  true 
solidity. 

232.  II.  Prismoidal  Formula. 

"  A  prismoid  is  a  solid  having  for  its  two  ends  any  dissimilar 
parallel  plane  figures  of  the  same  number  of  sides,  and  all  the 
sides  of  the  solid  plane  figures  also." 

Any  prismoid  may  be  resolved  into  prisms,  pyramids,  and 
wedges,  having  as  a  common  altitude  the  perpendicular  dis- 
tance between  the  two  parallel  end  planes. 

Let  AQ  and  AI  =  areas  of  end  planes. 

M  =  area  of  middle  section  parallel  to  the  end 

planes. 

I  =  length  of  prismoid,  or  perpendicular  dis- 
tance between  end  planes. 
S  =  solidity  of  the  prismoid. 
Then  it  may  be  shown  that 

l)1- 
o 


Methods  of  Computing  Earthwork.         131 

233.   Let  B  =  area  of  lower  face,  or  base  of  a  prism,  wedge, 
or  pyramid. 

b  =  area  of  upper  face. 

m  =  middle  area  parallel  to  upper  and  lower  faces. 
a  =  altitude  of  prism,  wedge,  or  pyramid. 
s  =  solidity  "      " 


"        " 


Then  the  area  of  the  upper  face  b  in  terms  of  lower  base  B 
will  be  for 


Prism 


Wedge 
6  =  0 


and  the  middle  area  m  will  be  for 
Prism  Wedge 

m=B  ro=| 

The  solidity  s  will  be  for  t 
Prism 


Pyramid 
6  =  0 


Pyramid 
m  =  - 


D 


O 


Wedge 


Pyramid 


Since  a  prismoid  is  composed  of  prisms,  wedges,  and  pyramids, 
the  same  expression  may  apply  to  the  prismoid,  and  this  may 
be  put  in  the  general  form 


(163  A) 


8  =  (Ao  +  4  M  +  A$ 
using  the  notation  of  the  preceding  page. 


132 


Railroad  Curves  and  Earthwork. 


234.  A  regular  section  of  earthwork  having  for  its  surface  a 
plane  face  is  a  prismoid.  Most  sections  of  earthwork  have  not 
their  surface  plane,  and  are  not  strictly  prismoids,  although 
they  are  so  regarded  by  some  writers. 

In  this  figure  the  lines  E000  and  EiOx  are  not  parallel,  and 
therefore  the  surface  00OiEiE0  is  not  a  plane.  The  most  com- 
mon assumption  as  to  this  surface  is  that  the  lines  000i  and  E0Ei 
are  right  lines,  and  that  the  surface  OoOiEiEo  is  a  warped  sur- 
face, generated  by  a  right  line  moving  as  a  generatrix  always 


.       G0         B0 

parallel  to  the  plane  00G0BoEo  and  upon  the  lines  000i  and 
E0Ei  as  directrices,  as  indicated  in  the  figure.  The  surface  thus 
generated  is  a  warped  surface  called  a  "hyperbolic  paraboloid." 
It  will  be  shown  that  the  "prismoidal  formula"  applies  also  to 
this  solid,  which  is  not,  however,  properly  a  prismoid. 

235.    In  the  following  figure,  which  has  perpendicular  sides 
L,  E0B0BiF     and  the  lines  D0E0  and  DiEi  right  lines, 

let  bo  =     base      =  A0B0 


" 


bl= 

c0  =  center  ht.  =  00G0 
_  D0Ao  +  E0Bo 

2 

d  =  center  ht.  =  OiGi 
=  DiAi  +  EiBi 

2 

I  =  length  (altitude) 

of  section  =  GoGi 

AQ  =  area  of  D0A0BoEo 

Ai  =  area  of 

S  =  solidity 


Methods  of  Computing  Earthwork.         133 

Also  use  notation  6X,  cz,  Ax  for  a  section  distant  x  from  GI. 
Then 


Cz  =  Ci  -  (Ci  -  C0)      =  Ci-f  (C0-  Ci) 


-  61)         ci  +  (c0  -  ci) 
+(c0  -  c^ 


^  =        7  (2  6iCi 
o 


(164) 


236.    Apply  the  "  Prismoidal  Formula  "  to  the  same  section. 
The  base  and  center  height  of  the  middle  section  are  :  — 


bm= 


r     _  Cp  + 

~ 


AQ  =  &oCo 

*=H 


_  area  Qf  middle  gection 


(165) 


This  is  the  same  as  formula  (164)  found  above  to  be  correct 
for  the  warped  surface.  Therefore  the  ' '  Prismoidal  Formula  " 
(163)  applies  to  the  section  shown  in  §  235. 


134         Railroad   Curves  and  Earthwork. 


237.  The  sections  of  earthwork  commonly  used  in  railroad 
work  are  bounded  not  by  perpendicular  sides,  but  by  inclined 

planes. 

In  the  figure,  suppose 
a  plane  to  be  passed 
through  the  line  EoEi, 
cutting  A0B0  at  P0  and 
AiBi  at  PI.  The  pris- 
moidal  formula  applies 
to  the  solid  E0PoBoBiEiPi 
cut  out  by  this  plane, 
since  this  solid  is  a  true 
prismoid.  If  the  pris- 
moidal  formula  applies 
to  the  entire  solid,  and 
also  to  the  part  cut  out, 
it  must  apply  to  the  re- 
maining solid  D0AoPoEoEiPiAiDi,  and  this  represents  in  form 
one  side  of  a  regular  three-level  section  of  earthwork  in  which 
DoAo  represents  the  center  height  and  EoPo  the  slope. 

If  the  prismoidal  formula  applies  to  the  section  upon  one  side 
of  the  center,  it  applies  also  to  the  other  side,  and  so  to  the 
entire  section. 

238.  The  "Prismoidal  Formula'1'1  is  of  wide  application. 
Since  it  applies  to  prisms,  wedges,  pyramids,   and   to   solids 
bounded  by  warped  surfaces  generated  as  described,  it  follows 
that  it  applies  to  any  solid  bounded  by  two  parallel  plane  faces 
and  defined  by  the  surfaces  generated  by  a  right  line  moving 
upon  the  perimeters  of  these  faces  as  directrices.     It  may  also 
be  stated  here  without  demonstration  that  it  also  applies  to  the 
frusta  of  all  solids  generated  by  the  revolution  of  a  conic  sec- 
tion as  well  as  to  the  complete  solids,  for  instance,  the  sphere. 

The  prismoidal  formula  is  generally  accepted  as  correct  for 
the  computation  of  earthwork  and  similar  solids,  and  the  meas- 
urements of  a  section  of  earthwork  are  taken  so  as  to 'repre- 
sent properly  the  surface  of  the  ground  if  this  be  a  warped 
surface  of  the  sort  described.  The  failure  to  use  the  prismoidal 
formula  is  explained  often  by  the  additional  labor  necessary 
for  its  use. 


Methods  of  Computing  Earthwork.         135 

239.  For  "three-level"  sections  of  earthwork,  a  result  cor- 
rect by  the  prismoidal  formula  may  be  secured,  and  the  work 
simplified,  by  calculating  the  quantities  first  by  the  inexact 
method  of  "end  areas,"  and  then  applying  a  correction  which 
we  may  call  "  The  Prismoidal  Correction." 

Let    Se  =  solidity  by  end  areas 

Sp=      "        "  prismoidal  formula 
Then  C  =  Se  —  Sp  =  prismoidal  correction 
In  the  figure,  §  235, 

Sp  =  by  formula  (164)  =  |  (2  bid  +  2  &0co  +  Mo  +  &(A) 

Se  =    (6ici  +  &0co)       =    (3  bid  +  3  &oc0) 


C=  Se  —  Sp  =-    (bid  +    &oCo  - 


'      =       (&i-&o)(ci-co) 

Let  D0A0  =  /io'  DiAi  =  ^' 

E0B0  =  hQ  EiBi  =  hi 


Then  c=l-(bi-  60)  (  hl  +  hl'  -h°  +  h()'} 

6  \       2  2       / 


=  ~(bi-b0)  (hi  +  W  -ho  -ho') 
129 


When  the  solid  assumes  a  trian- 
*   gular  cross-section,  as  in  the  figure, 


=0 


C  =(bi-bo)(hi-ho)    (166) 


136         Railroad  Curves  and  Earthwork. 


240.  If  any  solid  be  divided  into  a  number  ol  solids  each  of 
triangular  cross-section,  the  above  correction  may  be  applied  to 
each  such  triangular  solid,  and  the  sum  of  the  corrections  will 
be  the  correction  for  the  entire  solid. 


o, 


Let  this  figure  represent  a  section  of  earthwork  divided  into 
three  parts,  as  indicated  by  the  lines  D0G0,  E0G0,  DiGi,  EiGi. 

Then,  for  the  solid    00D0G0EoEiGiDiOi, 
C  =  -L  [(ci  -  c0)  (^  - 


ci  -  Co)  (^  -  e^0 


Let 


c^ 


and  D0  = 


For  the  solid  G0B0E0EiBiGi, 
(166)         C  = 


=  0 

Similarly  for  the  solid  A0G0D0DiGiAi. 
Hence  for  the  entire  solid 


Methods  of  Computing  Earthwork.         137 
When         I  =  100 


=  (d-Co)(Di-Z>o)incu.  yds.     (168) 


Since         C  =  8e-  8P 

Sp  =  Se-C  (169) 

When  (GI  —  c0)  (Z>i  —  Z>0)  is  positive,  the  correction  C  is  to 
be  subtracted  from  Se. 

When  (GI  —  c0)  (D\  —  DO}  is  negative,  the  arithmetical  value 
of  C  is  to  be  added  to  8e.  The  latter  case  seldom  occurs  in 
practice,  except  where  C  is  very  small,  perhaps  small  enough 
to  be  neglected. 

For  a  section  of  length  I, 


s*  =  ilo  ^'10°  ~  Cl00^ 

241.  In  general,  for  sections  of  earthwork,  the  prismoidal 
correction  as  given  above  applies  only  when  the  width  of  base 
is  the  same  at  both  ends  of  the  section.  There  are  certain 
special  cases,  however,  which  often  occur,  and  which  allow  of 
the  convenient  use  of  this  formula  for  prismoidal  correction. 
Referring  to  the  figure  on  p.  120,  and  the  corresponding  notes 
on  p.  118,  the  correction  can  be  correctly  applied  in  the  case  of 
the  excavation  from  Sta.  2  +  64  to  2  +  76  as  follows  :  — 

Compute  Se,  and  then  apply  C,  using  at 

Sta.  2  +  64  Do  =  23.2 

and  at  Sta.  2  +  76  DI  =  11.9  =  dfi 

or  the  distance  out  on  one  side  only.  This  may  readily  be 
demonstrated  to  be  proper  if  the  correction  to  the  right  of  the 
center  be  taken,  using  formula  (167),  and  the  correction  to  the 
left  using  formula  (166),  and  the  two  corrections  (right  and 
left)  be  added. 


138         Railroad  Curves  and  Earthwork. 

242.  Formula  (166)  can  also  be  used  to  find  the  correction 
for  the  triangular  pyramids  (for  excavation  Sta.  2  +  76  to  2  +  91, 
and  embankment  2  +  64  to  2  +  76),  each  end  of  the  pyramid 
being  considered  to  have  a  triangular  section.  A  much  simpler 
way  to  find  the  correction  for  a  pyramid  is  this, 

c=se-sp  =  ±se 

as  may  readily  be  shown  to  be  true  for  any  pyramid,  since 


P 
C  =  Se-Sp=Al-  =  ^  (170) 

243.  In  the  case  of  regular  "  Five-Level  Sections,"  as  shown 
in  the  figure,  p.  127,  the  prismoidal  correction  may  be  com- 
puted for  each  of  the  triangular  masses  bounded  by 

1.   AOB  2.    QBE  3.    OAD 

In  the  case  of  AOB,  the  prismoidal  correction  will  evidently 
be  =  0,  since  Z>0  =  0  =  DI,  and  therefore  DQ  —  DI  =  0. 
The  correction  for  the  mass  bounded  on  one  end  by 

QBE  =   C  =-^(/r0  -A)(dr0  -  drj 

and  by  OAD  =  C  =  ±  (flo  -  fj  (dl(>  -  dj 

iz 

OBE  and  OAD  differ  but  little  from  regular  sections  of  earthwork 
in  which  6  =  0. 

244.  In  the  case  of  "Irregular  Sections,"  the  prismoidal 
correction  cannot  with  convenience  be  accurately  employed. 
There  are,  however,  several  methods  by  which  we  may  calcu- 
late a  "prismoidal  correction"  which  will  be  approximately 
correct. 

For  the  purpose  only  of  calculating  the  correction,  either  of 
the  following  methods  may  be  employed  :  — 


Methods  of  Computing  Earthwork.         139 

1.  Neglect  all  intermediate  heights,   and  figure  correction 
from  center  and  side  heights. 

2.  Find  level  sections  of  equal  area  in  each  case,  and  figure 
correction,  using  the  center  heights  and  side  distances  of  these 
level  sections. 

3.  Having  c  and  D  of  the  irregular  section,  either 

(a)  retain  c  and  calculate  Z),  or 
(6)       "     D  "          "          c 

for  a  "  regular  three-level "  section  of  equal  area,  and  use  these 
values  to  calculate  the  correction. 

4.  Plat  the  cross-section  on  cross-section  paper,  and  equalize 
by  a  line  or  lines  drawn  in  the  most  advantageous  direction, 
and  from  the  c  and  D  thus  found  compute  the  correction. 

245.  In  relation  to  these  methods  :  — 

No.  1  is  most  rapid  and  least  accurate. 
"    2  is  less  exact  than  3  in  most  cases,  and  probably  no  more 

rapid. 
"    3  is  recommended  as  nearly  equal  to  4  in  accuracy,  and 

far  more  rapid. 
"    4  would  yield  the  most  accurate  results. 

The  value  of  these  approximate  methods  cannot  be  properly 
appreciated  until  certain  rapid  methods  of  computation  are 
understood,  as  will  be  appreciated  later. 

The  results  obtained  by  the  methods  shown  above  are  ap- 
proximate only,  but  in  most  cases  the  resulting  error  would  be 
small,  or  a  small  fraction  only  of  the  entire  correction,  which  is 
itself  generally  small. 

246.  The  method  of  calculating  by  averaging  end  areas 
and  applying  the  prismoidal  correction  will  be  found  much 
more  rapid  than  to  calculate  the  middle  area  and  apply  the 
prismoidal  formula  directly.      There  is  another  advantage  of 
importance  in  favor  of  the  use  of  the  prismoidal  correction  ; 
in  a  majority  of  cases  for  sections  used,  the  method  of  "'end 
areas,"  is  sufficiently  accurate  for  all  practical  purposes,  and 
from  the  use  of  the  prismoidal  correction  the  computer  will 
soon  learn  to  distinguish,  by  inspection  merely,  in  what  cases 
this  correction  need  be  applied. 


140          Railroad  Curves  and  Earthwork. 

247.  III.  Method  of  Middle  Areas. 

This  consists  in  calculating  the  area  of  the  middle  section 
(not  the  mean  of  the  end  areas),  and  assuming  the  solidity  to 
be  that  of  a  prism  having  a  base  equal  to  this  middle  area,  and 
an  altitude  equal  to  the  length  of  the  section  of  earthwork. 

Let  M  =  middle  area 

I  =  length  of  section 
Then  8  =  Ml 

This  method  is  not  exact.  It  gives  results  generally  less  than 
the  correct  solidity.  It  is  not  sufficiently  rapid  to  recommend  it. 

248.  IV.  Method  of  Equivalent  Level  Sections. 

This  consists  in  finding  level  end  sections  of  equal  area  with 
the  actual  end  sections  from  these  calculating  the  level  middle 
section,  assuming  the  top  surface  connecting  the  level  end  sec- 
tions to  be  a  plane ;  and  then  calculating  the  solidity  of  this 
prismoid  by  the  prismoidal  formula. 

This  method  is  not  exact ;  it  gives  results  less  than  the  correct 
solidity.  It  is  not  sufficiently  rapid  to  recommend  it. 

249.  V.  Method  of  Mean  Proportionals. 

This  consists  in  assuming  that  the  solid  is  the  frustum  of  a 
pyramid,  in  which  case  all  its  sides  would  meet  in  one  vertex. 
This  method  is  not  exact.  It  gives  results  always  less  than 
the  correct  solidity. 

250.  VI.  Henck's  Method. 

In  connection  with  the  prismoidal  formula,  it  was  stated  that 
the  most  common  assumption  was  that  the  upper  surface  is  a 
warped  surface  of  a  certain  kind  which  was  there  described. 
Henck's  Method  assumes  otherwise ;  that  the  upper  surface  is 
divided  into  plane  surfaces  by  diagonals  from  the  center  height 
of  one  cross-section  to  the  side  height  of  the  next,  as -shown 
in  the  figure,  where  the  diagonals  OiE0  and  00Di  are  drawn. 


Methods  of  Computing  Earthwork.         141 

Which  way  the  diagonals  shall  be  assumed  to  run  is  deter- 
mined on  the  ground  by  the  shape  of  the  surface  in  each  case. 


D, 


The  diagonals  OiEo  and  00Di  divide  the  surface  into  four  plane 
surfaces,  D000Di,  00DiOi,  00E0Oi,  E0OiEi. 

251.  Let  this  figure  represent  the  right-hand  side  of  a  section 
of  earthwork,  with  the  diagonal 
•assumed  to  run  from  E0  to  Oi. 

Join  EO  with  G0,  BI  and  GI. 

The  entire  solid  may  then  be 
considered  as  composed  of  three 
pyramids  having  their  vertices  in 
0  a  common  point  EQ. 

Using  notation  already  famil- 
iar, the  solidities  of  the  three 
pyramids  are  as  follows :  — 

si  =  area  G0B0BiGi  X  height  at  E0  -s-  3 


s2  =  area  G000OiGi  x  distance  out  to  E0  •*•  3 
^ 


2 
s3  =  area 


x  length  of  section  -s-  3 


142         Railroad  Curves  and  Earthwork. 


Let    Sr  =  solidity  of  this  right  half  of  section 
Sr  =  si  +  s2  +  SB 


Let     C  =  center  height  touched  by  diagonal 
H=  side         "  "         "        " 

D  =  distance  out  to  side  height  touched  by  diagonal 


Then  St=  ||"|  (fy,  +  ^  +  IT,)  +  c0^0  +  dd^  + 

=  solidity  for  left  half  of  section 
^=  ^r+  St=  |[|  (^r0  +  ^j0  +  ^  -f  ^  +  Hr  +  J3i) 

+  Ci  (^  +  dij  +  CrDr  +  a  A]  (HI) 

252.   An  example  will  further  show  the  application  of  this 
method. 


Sta. 

Surf. 
Elev. 

Grade 
Elev. 

Cross-Sections 

i 

inq  ft 

10  1  on 

13.0                 nQ                 II.  0 

+  3.0\        h2'°\       +1.0 
15.0     \                \       II.  0 

0 

123.0 

120.00 

+  5.0                                    +1.0 

The  notes  show  the  direction  of  the  diagonal  as  taken  on  the 
ground. 

In  this  case          6  =  20  s  =  1  to  1 


Methods  of  Computing  Earthwork.        143 
253.    Henck  gives  note-book  and  calculations  in  this  form  :  — 


STA. 

4 

i. 

c 

hr 

* 

dt+dr 

M*» 

/>r<?r 

AC, 

1 
0 

13.0 
15.0 

3.0\ 
5.0 
8.0 

2.0\ 
\3.0 

1.0 
\1.0 
8.0 

11.0 
11.0 

24.0 
26.0 

48.0 
78.0 
22.0 

22.0 

39.0 

4.0                           39.0 

14.0  x  —  =  140.0 

6)327.0 

5450  (cu.  ft.) 

254.   The  calculations  could  be  conveniently  made  however 
from  the  notes  as  now  generally  taken,  as  is  shown  below  :  — 


13.0 

+  3.0' 
15.0 

+  5.0 

8.0 


11.0 

+  1.0 

11.0 

+  1.0 

'  2.0 

8.0 

4.0 

14.0 


xf 


24.0  x  2.0  =    48.0 

26.0  x  3.0  =    78.0 

11.0  X  2.0=    22.0 
13.0  x  3.0  =    39.0 

=  140.0 


6)327.0 
5450  (cu.  ft.) 


255.  The  work  of  computation  would  not,  in  either  of  these 
cases,  properly  be  done  in  the  field  note-book,  but  rather  in 
a  calculation  book,  or  other  suitable  place. 

For  a  series  of  cross-sections,  Henck  systematizes  the  work, 
and  reduces  the  labor  noticeably  from  what  is  shown  here. 
(See  Henck's  Field  Book.)  Henck's  method  is  strictly  accu- 
rate, upon  the  assumption  made  as  to  the  upper  surface.  In 
general  railroad  practice,  most  engineers  prefer  to  assume  the 
upper  surface  a  warped  surface  of  the  sort  described. 

Henck's  method  is  less  rapid  than  that  of  averaging  end  areas 
and  applying  the  prismoidal  correction. 


144 


Railroad  Curves  and  Earthwork. 


256.  The  method  of  averaging  end  areas  and  applying  the 
prismoidal  correction  appears  in  point  of  accuracy  and  rapidity 
to  meet  the  requirements  of  modern  railroad  practice. 

Some  engineers  whose  opinions  are  entitled  to  careful  con- 
sideration object  to  the  use  of  the  prismoidal  formula  or 
prismoidal  correction  in  any  form,  some  as  an  unnecessary 
refinement,  and  some  on  the  ground  that  certain  practical  con- 
siderations render  the  results  nearer  the  truth  when  the  method 
of  averaging  end  areas  is  used  without  applying  the  prismoidal 
correction.  Probably  the  greater  part  of  the  best  engineering 
practice  favors  the  use  of  the  prismoidal  correction. 

257.  Example. 

Showing  a  comparison  of  various  methods  of  calculating 
earthwork. 
Notes  of  excavation.     Base  24.    Slope  1|  to  1. 


Area  of  grade  triangle, 


24  x  8 


Sta.  1. 


2 

9  x30 


=  96 


+  3.0 


=  135 


Grade  triangle  =    96 


Sta.  0. 


+  11.0  +  21.0 

The  mid-section  will  be 


Grade  triangle'  =    96 
Ao      =  876 


Grade  triangle  =   96 
=  363 


258.   I.   End  Areas. 


C  =  iff.  x  18  x  42 
Error  of  8,  =  +  6300  =  16  per  cent 


=  6300 
=  39450 


Methods  of  Computing  Earthwork.         145 


259.  II.   Prismoidal  Formula. 

39      =Ai 
363  X  4  =  1452      =  4  AQ+SO 

876      =  AQ 
6)236700 

39450  =  Sp 

260.  III.  Middle  Areas. 

Sm  =  363  x  100  =    36300 

Sm  error  =  -  3150  =  8  per  cent 
Se       "     =  +  6300 

261.  IV.   Equivalent  Level  Sections. 


I  =  Area  of  level  section 
Al  =  \\  e2 


An 


ezo  —  135  (this  includes  grade  triangle) 

eV=90 

e0  =9.5± 

e2i  =  972 

e\  =  648 

ei  =25.5± 


ew=  25,5^5  = 

2 

Atm=  IJ  X  17.5a  =  457.8 

Grade  triangle  =    96.0 

361.8 

39.      =  Ai 
4x361.8  =  1447.2    =  4  Am 

876.      =  ^lo 
6)2362.20 
39370  =  Si 


Error  =  -  80 

=  0.2  per  cent 


146 


Railroad  Curves  and  Earthwork. 


262.  V.   Mean  Proportionals. 

Ai=       39  39.      -AQ 

,=      876  184.8    =VA^ 

=  34164  876.      =  Al 

=      i84.8      3)1099.80 

36660.  = 

Error  -  2790.  =  7  per  cent 

263.  VI.  Henck's  Method. 

ONE  SYSTEM  OF  DIAGONALS  AS  SHOWN  IN  NOTES. 


STA. 

di 

ft, 

c 

*, 

fc 

«+i 

(di+  dr)c 

D'C' 

7 

2. 

1 

0 

13.5 

28.5 

1.0 

11.  oy 

12.0 

/1.0\ 
19.0 

6. 

a 

3.0 
\21.0 
24.0 
12.0 
11.0 
21.0 
68.0 
=    12. 

16.5 
43.5 

8, 

30.0 
72.0 

6 

-  sk  -- 

30.0 

1368.0 
72.0 
816.0 
)2286. 

38100  =  8h 
=  +  1350 

816. 


=  3  per  cent 


OPPOSITE  SYSTEM  OF  DIAGONALS. 


STA. 

di 

hi 

c 

?>r 

dr 

di  +  dr 

((?z  +dr)c 

DC' 

57 

DC 

1 
0 

13.5 

28.5 

1.0\ 
11.0 

1.0 
\19.0/ 

/3.0 
21.0 

16.5 
43.5 

30.0 
72.0 

30.0 
1368.0 

0. 

24.0 
12.0 
1.0 
3.0 
40.0 
*  =    12. 
2      480. 

6^  -  /S»'  =  -  1350  =  3  per  cent 


570.0 

480.0 

6)2448.0 

40800  =  8h> 
38100  =  Sk 
2)78900 
39450 
Mean  value  =  S0 


CHAPTER   XIII. 
SPECIAL  PROBLEMS. 

264.   Correction  for  Curvature. 

In  the  case  of  a  curve,  the  ends  of  a  section  of  earthwork  are 
not  parallel,  but  are  in  each  case  normal  to  the  curve.  In  cal- 
culating the  solidity  of  a  section  of  earthwork,  we  have  hereto- 
fore assumed  the  ends  parallel,  and  for  curves  this  is  equivalent 
to  taking  them  perpendicular  to  the  chord  of  the  curve  between 
the  two  stations. 

Then,  as  shown  in  Fig.  1  (where  IG  and  GT  are  center-line 
chords),  the  solidity  (as  above)  of  the  sections  IG  and  GT  will 
be  too  great  by  the  wedge-shaped  mass  RGP,  and  too  small  by 


FIG.  1. 

QGS.  When  the  cross-sections  on  each  side  of  the  center 
are  equal,  these  masses  balance  each  other.  When  the  cross- 
section  on  one  side  differs  much  in  area  from  that  on  the  other, 
the  correction  necessary  may  be  considerable. 

147 


148         Railroad   Curves  and  Earthwork. 


In  Fig.  2,  use  c,  hi,  hr,  dt. 
Let  D  =  degree  of  curve. 

Q     E     S 


dr,  6,  s,  as  before. 

Make  BL  =  AD,  and  join  OL. 

Then    ODAG    balances 
OLBG,  and  there  remains 
an  unbalanced  area  OLE. 
Draw  OKP  parallel  to 
AB. 

By  the  "Theorem  of 
Pappus  "  (see  Lanza,  Ap- 
plied Mechanics),  "If  a  plane 
area  lying  wholly  on  the  same 
side  of  a  straight  line  in  its  own 
plane  revolves  about  that  line, 
and  thereby  generates  a  solid 
of  revolution,  the  volume  of  the 
solid  thus  generated  is  equal  to 
the  product  of  the  revolving 
area  and  of  the  path  described  by  the  center  of  gravity  of  the 
plane  area  during  the  revolution." 

The  correction  for  curvature,  or  the  solidity,  developed  by 
this  triangle  OLE  (Fig.  2)  revolving  about  OG  as 
an  axis  will  be  its  area  x  the  distance  described 
by  its  center  of  gravity.     The  distance 
out  (horizontal)  to  the 
center  of  gravity  from 
the  axis  (center  line) 
will  be  two  thirds  of 
the  mean  of  the  dis- 
tances out  to  E  and  to 
L,or 

=  3         2 
and  the  distance  described  will  be 


FIG.  2. 


The  area 


OLE  = 


Special  Problems.  149 

Therefore  the  correction  for  curvature, 

AJ  .  ^         x  angle  QGS 


When  IG,  GT  are  each  a  full  station,  or  100  ft.  in  length, 
QGS  =  D 

\  •  ftr-fr.£±*x  angle  D 
/          2 


2 
arc  1°  =  .01745 

x  0.01745  Z) 


=  f±  +  8C\  hr-hlxdr  +  dl 

\  (hr  -  hi)  (dr  +  d{)  x  0.00291  D  (cu.  ft.)          (172) 
(hr  -  h{)  (dr  +  di)  x  0.00011  D  (cu.  yds.)        (173) 


=•     +  sc 


265.    When  IG  or  GT,  or  both,  are  less  than  100  ft.,  let 
IG  =  Z0         -and         GT  =  h 

Then  QGE=Ax|andSGE=!|jxf 


200 


266.   The  correction  C  is  to  be  added  when  the  greater  area 
on  the  outside  of  the  curve,  and  subtracted  when  the  greater 
area  is  on  the  inside  of  the  curve. 
.^     When  the  center  height  is  0,  as 
^s^/       in  Fig.  3,  we  may  consider  this 
/  a  regular  section  in  which  c  =  0, 

B  hi  =  0,  and  di  =  - :  then 


,  =    ;  then 

FIG.  3. 


x  0.00011  D  (cu.  yds.)  (175) 


150         Railroad   Curves  and  Earthwork. 


In  the  case  of  an  irregular  section,  as  shown  in  Fig.  4,  the 
area  and  distance  to  center  of  gravity  (for  example,  of  OH  EM  L) 
may  be  found  by  any  method  available,  and  the  correction 


FIG.  4. 

figured  accordingly.  The  correction  for  curvature  is,  in  present 
railroad  practice,  more  frequently  neglected  than  used.  Never- 
theless, its  amount  is  sufficient  in  many  cases  to  fully  warrant 
its  use. 

267.    Opening  in  Embankment. 

Where  an  opening  is  left  in  an  embankment,  there  remains 
outside  the  regular  sections  the  mass  DEKHF. 

D 


This  must  be  calculated  in  3  pieces,  ADF,  BEKH,  ABHF. 
Let  b  =  base  =  AB 

dr  =  distance  out  right 

di  =  distance  out  left 

Pr=BH 

Pi 


|  taken  parallel  to  center  line 


•£l}heightsat{AB 

«i  =  solidity  ADF 
s2  =  BEKH 
s3  =  ABHF 


Special  Problems.  151 

Then  (approximately)  following  the  "Theorem  of  Pappus," 

s\  =  mean  of  triangles  AD  and  AF  x  distance  described  by 
center  of  gravity. 


area  AD  +  area  AF 
mean  area  = 


The  distance  described  by  center  of  gravity  is  found  thus  :  — 

length  AD  =  di  -  -,  and  AF  =  pt 
mean  length  =  length  AD  +  length  AF 


distance  out  to  center  of  gravity  =  -  .  - 1  dt  +  PI  —  -  ) 
>  3     2  \  2  / 

distance  described  by  center  of  gravity  =  —  (  di  +  pi—-\  — 

6\  2/2 


52  =        dr  +Pr  ~ 


_  area  AF  -f  area  BH      AD 

03  — —  X  AD 


152          Railroad   Curves  and  Earthwork. 

The  work  of  deriving  formulas  (176)  and  (177)  is  approxi- 
mate throughout,  but  the  total  quantities  involved  are  in  gen- 
eral not  large,  and  the  error  resulting  would  be  unimportant. 

There  seems  to  be  no  method  of  accurately  computing  this 
solidity  which  is  adapted  to  general  railroad  practice. 

268.  Borrow-Pits. 

In  addition  to  the  ordinary  work  of  excavation  and  embank- 
ment for  railroads,  earth  is  often  "borrowed"  from  outside 
the  limits  of  the  work  proper  ^  and  in  such  excavations  called 
"borrow-pits,"  it  is  common  to  prepare  the  work  by  dividing 
the  surface  into  squares,  rectangles,  or  triangles,  taking  levels 
at  every  corner  upon  the  original  surface  ;  again,  after  the 
excavation  of  the  borrow-pit  is  completed,  the  points  are  repro- 
duced and  levels  taken  a  second  time.  The  excavation  is  thus 
divided  into  a  series  of  vertical  prisms  having  square,  rectangu- 
lar, or  triangular  cross-sections.  These  prisms  are  commonly 
truncated  top  and  bottom.  The  lengths  or  altitudes  of  the 
vertical  edges  of  these  prisms  are  given  by  the  difference  in 
levels  taken, 

1st,  on  the  original  surface,  and 

2d,  after  the  excavation  is  completed. 

This  method  of  measurement  is  very  generally  used,  and  for 
many  purposes. 

269.  Truncated  Triangular  Prisms. 

K  Let  A  =  area  of  right  section  EFD  of  a 

truncated  prism,  the  base  ABC 
being  a  right  section 

hi  =  height  AH 
h2  =  «  BE 
hs  =  "  CK 

a  =  altitude  of  triangle  EFD  dropped 
from  E  to  FD 

Let  S  =  solidity  of  prism      ABCKHE 

st  =      "        "      "  ABCFDE 

su  =      "        "  pyramid  FDEHK 


Special  Problems. 


153 


Then 


su  =  area  DFKH  x  - 
3 


8  =  $i  +  su  =  A I 


AD  +  BE  +  CF 


**) 


fl2  + 


(179) 


If  the  prism  be  truncated  top  and  bottom,  the  same  reason- 
ing holds  and  the  same  formula  applies. 

270.   Truncated  Rectangular  Prism. 

Let  A  =  area  of  right  section  A  BCD 
of  a  rectangular  prism 
truncated  on  top  (base 
is  ABCD) 

hi  =  height  AE 
hz  =  "  BG 
hs  -  "  KC 
h=  "  HD 

S  =  solidity  of  prism 

b  =  AD  =  BC 

a  =  AB  =  DC 


154         Railroad  Curves  and  Earthwork. 

Then  using  method  of  end  areas, 

0_ 
fifs 


2 

xa 


<  > 

"We  may  find  £,  correct  by  the  prismoidal  formula,  if  we 
apply  the  prismoidal  correction.  The  prismoidal  correction 
(7  =  0,  since  D0  -  D\  =  0  (or  in  this  case  AD  -  BC  =  0).  The 
formula  therefore  remains  unchanged.  It  is  evident  from  this, 
then,  that  the  solution  holds  good,  and  the  formula  is  correct, 
not  only  when  the  surface  EH  KG  is  a  plane,  but  also  when  it  is 
a  warped  surface  generated  by  a  right  line  moving  always  par- 
allel to  the  plane  ADHE,  and  upon  EG  and  HK  as  directrices. 

Some  engineers  prefer  to  cross-section  in  rectangles  of  base 
15'  x  18'.  In  this  case 


(cu>     dg 


27  4 

=  10  **  +  h*  +  hs  +  ^4  (cu.  yds.)  (182) 

Other  convenient  dimensions  will  suggest  themselves,  as 
10'  x  13.5'     or    20'  x  13.5'     or    20'  x  27' 

By  this  method  the  computations  are  rendered  slightly  more 
convenient;  but  the  size  of  the  cross-section,  and  the  shape, 
whether  square  or  rectangular,  should  depend  on  the  topog- 
raphy. The  first  essential  is  accuracy  in  results,  the  second 
is  simplicity  and  economy  in  field-work,  and  ease  of  computation 
should  be  subordinate  to  both  of  these  considerations. 


Special  Problems. 


155 


271.   Assembled  Prisms. 

In  the  case  of  an  assembly  of  prisms  of  equal  base,  it  is  not 
necessary  to  separately  calculate  each  prism,  but  the  solidity  of 
a  number  of  prisms  may  'be  calculated  in  one  operation. 

In  the  prism  B, 

S    =  A  az  +  as  "*"  &3  +  ^2 


From  inspection  it  will  be  seen,  taking  A  as  the  common 
area  of  base  of  a  single  prism,  and  taking  the  sum  of  the 
solidities,  that  the  heights  a^,  a&  enter  into  the  calculation  of 


B 

C 

D 

b, 

b2 

b3 

b. 

b5                 6 

E 

F 

G 

H 

I 

Ci 

C2 

C3 

c, 

C5 

K 

L 

M 

N 

one  prism  only  ;  «3,  «4  into  two  prisms  each  ;   61,  66  one  only  ; 
&2,  &5  into  three  prisms  ;  63,  &4  into  four  prisms  ;  and  similarly 
throughout. 
Let      t\  =  sum  of  heights  common  to  one  prism 

£2  =    "     "      "  "          "          two  prisms 

t3=    "     "      "  *         "          "          three    " 
"      "  "          "          four     «• 


Then  the  total  solidity, 


.  yd8i) 


(183) 
(184) 


156 


Railroad  Curves  and  Earthwork. 


272.    Additional  Heights. 

When  the  surface  of  the  ground  is  rough  it  is  not  unusual 
to  take  additional  heights,  the  use  of  which,  in  general,  involves 
appreciable  labor  in  computation,  it  being  necessary  commonly 
to  divide  the  solid  into  triangular  prisms,  as  suggested  by  the 
figures  just  below,  which  include  the  case  of  a  trapezoid. 


\                       / 

^""x<^ 

xx              \ 

\  / 

'                            \ 

f                                                        \ 

/                       \ 

v^y 

The  computations  may  be  simplified  in  the  two  special  cases 
which  follow : 

(a)    When  the  additional  height  hc  is 
in  the  center  of  the  rectangle. 

Here   the   solid   is   composed   of   an 
assembly  of  4  triangular  prisms  whose 

right  sections  are  of  equal  area  =  —  • 


The  solidity  of  the  assembled  prisms 


j     - 
3 


or  the  total  solidity  is  that  due  to  the  four  corner  heights  plus 
the  solidity  of  a  pyramid  of  equal  area  of  base  and  whose  alti- 
tude is  the  difference  between  the  center  height  and  the  mean 
of  the  four  corner  heights. 


Special  Problems. 


157 


(6)  When  the  additional  height  is  at 
the  middle  of  one  side  of  the  rectangle.     hf 


=       (3  ft 


hi 


or  the  total  solidity  is  that  due  to  the  four  corner  heights  plus 
the  solidity  of  a  pyramid  of.  equal  area  of  base  and  whose  alti- 
tude is  the  difference  between  the  middle  height  and  the  mean 
of  the  adjacent  side  heights. 

Apparently  the  principle  of  the  pyramid  applies  conveniently 
only  in  these  two  cases. 

For  the  case  where  the  point  lies  on  one  of  the  sides,  an 
alternate  method  of  dividing  the  rectangle  (or  trapezoid)  is 
indicated  below. 


The  details  of  the  computation  in  this  case  need  not  be 
worked  out  here. 


CHAPTER   XIV. 


EARTHWORK  TABLES. 

273.  The  calculation  of  quantities  can  be  much  facilitated 
by  the  use  of  suitably  arranged  "Earthwork  Tables." 

For  regular  "Three- Level  Sections"  very  convenient  tables 
can  be  calculated  upon  the  following  principles  or  formulas  :  — 


A  G  B 

Use  notation  as  before  for 

c,  hh  hr,  dh  dr,  s,  Z,  A,  S 
Then        A  =     ABKL     +         OKE         -     DDL 

OK  x  EM         OLx  ND 
2  2 

°K(EM-ND) 


=  c(6  +  sc)  + 


A  =  c(b  +  sc)  + 1  (|  +  sc\  (hi  +  hr-2c) 
For  a  prism  of  base  A  and  I  —  50, 


S  =  50  A  (cu.  ft.)  =  —  A  (cu.  yds.) 


158 


^Sc)  (cu.yds.)     (186) 


Earthwork  Tables. 


159 


274.  For  cross-sections  of  a  given  base  and  slope,  that  is, 
given  b  and  s  constant,  we  may  calculate  for  successive  values 
of  c,  and  tabulate  values  of  L  and  K  as  follows :  — 


L  represents  the  solidity  for  the  level  section. 
K  is  for  use  as  a  correction.    The  formula  then  adapts  itself 
to  this  table  for  any  desired  values  of  c,  hi,  hr. 

S  =  L  +  K(hi  +  hr  —  2  c)  (186  A) 

Having  found  for  successive  stations  So  and  Si  (each  for  a 
prism  Z  =  50),  then  for  the  full  section  by  "  end  areas," 


for 


$100  =  So  +  Si 

o  AQ  +  AI 

moo  = 


100  =  50.Ao 

27         27  27 


Seioo  — 
275.   When  I  is  less  than  100, 

«.  =  («>  + A)  j^ 
For  level  sections  hi  —  . 


(187) 


(188) 


and  the  formula 


becomes 


S  =  L 


(189) 


for  level  sections,  and  the  quantities  for  any  given  values  of  c 
can  be  directly  taken  from  column  L  without  any  correction 
from  column  K. 

In  preliminary  estimates,  or  wherever  center  heights  only 
are  used,  such  tables  are  rapidly  used. 


160         Railroad  Curves  and  Earthwork. 

276.   Tables  may  be  found  at  the  back  of  the  book,  pages 
190,  191,  calculated  for 

1.  6  =  20  s  =  l|tol 

2.  6  =  14  s  =  1£  to  1 

An  example  will  illustrate  their  use, 

b  =  14  s  =  1£  to  1 

Notes :  — 

Sta.  I  — ^  -3.7  — 

-4.0  -3.6 

10.6  10.3 


-  2.4  -  2.2 

Calculations  :  — 
3.7 


2.5 


£100  = 

277.   There  is  also  at  the  back  of  this  book  a  "Table  of 
Prismoidal  Correction  "  calculated  by  the  formula 


L  =  134.0 
+      2.3 
Si  =  136.3 

£•=11.6 

0.2^- 
2.32 

ht  +  hr  =  7.6 
^^  2c  =  7.4 

"^^^+0.2 

L=    82.2 
-      4.0 

S0=    78.2 

K=  10.0 
0.4^- 
4.00 

hi+hr  =  4.6 
>^^  2  c  =  5.0 
"^^.-0.4 

#o  =  214.5 

In  the  example  above  . 

c0  -  ci  =   2.5-    3.7  =-1.2 
D0-Dl=  20.9  -  25.4  =  -  4.5 

From  Table  find  opp.  4.5  for  1  1.39 

2.78(10 

—          0.28     -     0.28 
10    C  =  L67 


C=     1.7 

Sp  =  212.8 


Earthwork  Tables.  161 

278.    When  the  section  is  less  than  100  ft.  in  length,  the 
prismoidal   correction    is    made   before   multiplying  by  —  ; 

1UU 

that  is,  spl  =  (SQ+8l-C}^-  (190) 


279.  Tables  based  upon  these  formulas  have  been  published 
as  follows  :  — 

"  The  Civil  Engineer's  Excavation  and  Embankment  Tables," 
by  Clarence  Pullen  and  Charles  C.  Chandler,  published  by  the 
"  J.  M..W.  Jones  Stationery  and  Printing  Co.,"  Chicago. 

Tables  are  calculated  for  b  =  12,     14,     16,     18,     20 

s=    i,       i,       1,     H 
Tables  of  a  similar  kind,  but  calculated  so  that 

r»  Sl-j-So 

moo  =  -  -  — 

are  Hudson's  Tables,  published  by  John  Wiley  &  Sons,  New 
York. 

280.  For  general  calculation  adapted  both  to  regular  «  '  Three- 
Level  Sections"  and  to  "Irregular  Sections,"  tables  can  be  cal- 
culated upon  the  following  principles  and  formulas  :  — 

These  tables  are,  in  effect,  tables  of  "Triangular  Prisms,"  in 
which,  having  given  (in  feet)  the  base  B  and  altitude  a  of  any 
triangle,  the  tables  give  the  solidity  (in  cu.  yds.)  for  a  prism  of 
length  I  =  50  ;  that  is, 


Whenever  the  calculation  can  be  brought  into  the  form 
8  =  77  «5,  the  result  can  be  taken  directly  from  the  table. 

54 

281.  Tables  of  this  kind  are  "Allen's  Tables  for  Earthwork 
Computation,"  by  the  author  of  this  book.  A  sample  page  is 
shown  at  the  end  of  this  book,  page  198.  Convenient  tables  of 
the  same  kind,  but  arranged  differently  for  use,  are  "Tables 
for  the  Computation  of  Kail  way  and  other  Earthwork,"  by 
C.  L.  Crandall,  C.E.,  Ithaca,  New  York. 


162          Railroad  Curves  and  Earthwork. 


282.    In  both  tables  the  formula  S  =  —  aB  takes  form  thus, 

50  54 

S  =  —  x  width  x  height,  and  the  tables  are  arranged  as  below. 
54 


HEIGHTS. 

WIDTHS 

—  width  x  height 
54 

The  application  to  "  Three-Level  Sections  "  is  as  follows 
We  have  formula  (162),  p.  126, 


and  for  a  prism  50  ft.  in  length  (I  =  50) 

---»     (192) 


or  S  is  the  sum  of  two  quantities,  each  of  which  is  in  proper 
*  form  for  the  use  of  the  tables. 

For  cross-sections  of  a  given  base  and  slope  (6  and  s  con- 
stants), —  is  a  constant,  and  also  —  • b  is  constant. 

2s  54    2  s 

We  may  then  calculate  once  for  all  - — ,  and  call  this  B  (a 
constant). 

Also  —  .  —  •  6,  and  call  this  a  constant  E. 
54    2s 

Cf\ 

Then  S  = 


In  using  the  tables,       c  +  B  =  height 
D  =  width 
As  in  the  previous  tables,  having  found  So  and  Si, 

$100  =  $0  +  Sl 

and  fl=(ff0  +&)_!- 


Earthwork  Tables.  163 

283.  Example.    Allen's  Tables  for  Earthwork  Computation. 

Notes:  — 

Sta.  I  -M  -1.2 

-2.4  -  1.2 


6  =  11  s  =  11  to  1 

A  =  3.7  =  B 

2s 

Grade  triangle,  —  x  3.7  x  11 

54 

Under  height  3.7,  find  1  =    3.43  10.    =  34.3 

1=    3.43  1.    =    3.4 

JJT   OIT     n 

Station  1.  c  =    1.2 

B=    3.7 

height  =  ~19 

Z>  =  9.1  +  7.3  =  16.4 

Under  height  4.9,  find  1  =    4.54  10.    =  45.4 

6  =  27.22  6.    =  27.2 

4  =  18.15  .4=    1.8 

74~4 
.£7  =  37.7 

Si  =  36/7 
Station  0.  c=    0.7 

B=   3.7 

height  =   4.4 
D  =  8.8  +  6.4  =  15.2 

Under  height  4.4,  find  1=    4.07  10.    =40.7 

5  =  20.37  5.    =  20.4 
-      2=    8.15  .2  =    0.8 

61.9 
J£  =  37.7 


164          Railroad  Curves  and  Earthwork. 
284.   Irregular  Sections. 


P        B 


An  "Irregular  Section"  can  be  divided  into  triangular  parts, 
as  in  the  figure.  Taking  generally  two  triangular  parts  together 
for  purposes  of  calculation,  we  have 


Ai  -hlX  (AG  ~ 
2 

A-2  =• 

A3  =  J — 
^4=  — 


-  Q) 


A    _hpx  (dB  - 

~~ 


(dr  - 


54 


,  N 
-c?H) 


85  -f 


(193) 


285.  The  calculation  of  Irregular  Sections  in  rough  country 
becomes  very  laborious  unless  the  best  methods  are  used,  and 
this  process  should  be  thoroughly  understood. 


CHAPTER  XV. 


EARTHWORK  DIAGRAMS. 

286.  Computations  of  earthwork  may  also  be  made  by  means 
of  diagrams  from  which  results  may  be  read  by  inspection 
merely. 

The  principle  of  their  construction  is  explained  as  follows  :  — 
Given  an  equation  containing  three  variable  quantities  as 


x  =  zy 


(194) 


If  we  assume  some  value  of  z  (making  z  a  constant),  the 

equation  then  becomes  the  equation  of  a  right  line. 
If  this  line  be  platted,  using  rectangular  coordinates  (as  the 

line  z  —  1  in  the  figure) ,  then  having  given  any  value  of  y,  the 
corresponding  value  of  x  may  be 
taken  off  by  scale.  If  a  new  value 
of  z  be  assumed,  the  equation  is 
obtained  of  a  new  line  which  may 
also  be  platted  (as  z=.\  in  the 
figure),  and  from  which  also,  hav- 
ing given  any  value  of  y,  the  cor- 
responding value  of  x  may  be 
determined  by  scale.  Assuming 
a  series  of  values  of  z  and  platting, 
we  have  a  series  of  lines,  each 
representing  a  different  value  of  z, 

and  from  any  one  of  which,  having  given  a  value  of  y,  we  may 

by  scale  determine  the  value  of  x. 
Thus,  given,  values  of  z  and  y  ;  required,   x,  we  may  find, 

1.  The  line  corresponding  to  the  given  value  of  z,  and 

2.  Upon  this  line  we  may  find  the  value  of  x  corresponding 
to  the  given  value  of  y. 

165 


166          Railroad  Curves  and  Earthwork. 


- 

/ 

/ 

/ 

K 

y 

^" 

/ 

c< 

h^* 

^ 

/ 

^ 

/• 

^^ 

a    i 

4     5     a 

i 

287.  Next,  if  instead  of  platting  upon  lines  as  coordinate 
axes,  we  plat  upon  cross-section 
paper,  the  cross-section  lines  form 
a  scale,  so  that  the  values  of  x  and 
y  need  not  be  scaled,  but  may  be 
read  by  simple  inspection  as  in  the 
figure. 

288.    If  the  equation  be  in  the 
form 

x  =  azy  (195) 

the  same  procedure  is  equally  possible,  and  the  line  represent- 
ing any  value  of  z  will  still  be  a  right  line. 
If  the  equation  be  in  the  form 

x  =  a(z  +  6)  (y  +  c)  +  d  (196) 

in  which  a,  6,  c,  d,  are  constants,  the  same  procedure  is  still 
possible,  and  the  line  representing  a  given  value  of  z  is  a  right 
line,  as  before. 

The  use  of  diagrams  of  this  sort  is  therefore  possible  for  the 
solution  of  equations  in  the  form  of 


or  in  simpler  modifications  of  this  form. 

289.  Referring  again  to  the  figure  above,  we  may  consider 
the  horizontal  lines  to  represent  successive  values  of  x  and  refer 
to  them  as  the  lines 

x  =  Q  ;  x  =  l;  x  =  2,  etc. 
and  similarly  we  may  refer  to  vertical  lines  as  the  lines 

y  =  0;  y  =  l;  y  =  2,  etc. 
just  as  we  refer  to  the  inclined  lines 

z  =  \  ;  2  =  1,  etc. 

Having  given  any  two  of  the  quantities  x,  ?/,  z,  the  third  may 
be  found  by  inspection  from  the  diagram  by  a  process  similar 
to  that  described. 


Earthwork  Diagrams.  167 

290=   Diagram  for  Prismoidal  Correction. 
Formula  c  =  _L  (c0  -  ci)  (Z>0  -  Z>i)  (168) 

o.^s4 

This  has  the  form    x  =    a  x      £      X       y 

Construction  of  diagram. 

Assume  (as  we  did  for  z~)  a  series  of  values  of 

c0  -  ci  =  0,  1,  2,  3,  4,  5,  etc. 

When  c0  -  ci  =  0    then     C  =  0 
or,  the  line  c0  —  Ci  coincides  with  the  line  (7  =  0. 

When  Co  —  Ci  =  1,  the  equation  of  the  line  c0  —  c\  is 

°  =  m^D9~D° 

To  plat  this  right  line,  we  must  find  two  or  more  points  on 
the  line.  For  the  reason  that  cross-section  paper  is  generally 
warped  somewhat,  it  is  best  to  take  a  number  of  points  not 
more  than  3  or  4  inches  apart,  in  order  to  get  the  lines  suffi- 
ciently exact.  For  convenience,  take  values  of  DO  —  D\  as 
follows  :  — 

When  (c0  —  ci)  =  1 

take   A>-  £>i  =  0,    3.24,    6.48,    9.72,    12.96,    16.20,  etc. 
then  0  =  0,      1,      2,      3,       4,        5,  etc.  ' 

When  CQ  —  Ci  =  2,  the  equation  of  the  line  c0  —  Ci  is 


Therefore  when  c0  —  ci  =  2 

take   #o-#i  =  0,    3.24,    6.48,    9.72,    12.96,    16.20,  etc. 
then  0  =  0,      2  ,      4  ,      6  ,       8    ,      10  ,  etc. 


168 


Railroad  Curves  and  Earthwork. 


291.    In  like  manner  a  table  may  be  constructed. 


0 

3.24 

6.48 

9.72 

12.96 

16.20 

19.44 

22.68 

26.92 

*-A 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

2 

0 

2 

4 

6 

8 

10 

12 

14 

16 

3 

0 

3 

6 

9 

12 

15 

18 

21 

24 

4 

0 

4 

8 

12 

16 

20 

24 

28 

32 

5 

0 

5 

10 

15 

20 

25 

30 

35 

40 

6 

0 

6 

12 

18 

24 

30 

36 

42 

48 

7 

0 

7 

14 

21 

28 

35 

42 

49 

56 

8 

0 

8 

16 

24 

32 

40 

48 

56 

64 

9 

0 

9 

18 

27 

36 

45 

54 

63 

72 

10 

0 

10 

20 

30 

40 

50 

60 

70 

80 

*-,, 

292.    It  will  be  noticed  that  when  D0  —  I>\  —  0,  (7  =  0. 

Therefore  for  all  values  of  c0  —  Ci,  the  lines  pass  through  the 
origin. 

We  may  proceed  to  plat  the  lines  c0  —  GI  =  1,  c0  —  Ci  =  2, 
C0  —  GI  =  3,  etc.,  from  data  shown  in  the  above  table,  platting 
upon  the  lines  Z>0  —  D\  =  3.24,  Z>0  —  DI  =  6.48,  etc.,  the  points 
shown  with  circles  around  them  in  the  cross-section  sheet, 
p.  165. 

Having  the  lines  c0  —  Ci  =  1,  c0  —  Ci  =  2,  3,  platted,  inter- 
mediate lines  are  interpolated  mechanically  upon  the  prin- 
ciple that  vertical  lines  would  be  proportionally  divided  (as 
ML  is  proportionally  divided  into  5  equal  parts),  and  points 
are  marked  for  the  lines 

C0-ci  =  1.2,         1.4,        1.6,         1.8 

For  the  most  convenient  use,  the  values  of  c0  —  Ci  are  taken 
to  every  second  tenth  of  a  foot  in  interpolating,  as  is  shown  on 
the  diagram,  p.  165,  between  1  and  2  ;  that  is, 

1.2,  1.4,  1.6,  1.8 

A  complete  diagram  is  shown  at  the  back  of  the  book. 


Earthwork  Diagrams. 


169- 


0                                                CP                                                  0                                                  £?                            0 

sr 

V 

i\ 

] 

\\ 

\ 

f 

u 

Of 

3.21 

71 
CO 

V 

s 

s 

Q 

•V 

l\\ 

\ 

s 

s 

£ 

33 

tY\ 

3 

f_- 

—  Ss 

(f0- 

Dl= 

5.48 

O 

o 

\\ 

\v 

\\ 

^ 

\ 

33 
m 
-O 

\ 

\\ 

\\ 

\ 

\ 

0 

E 

t\- 

\ 

vs 

\ 

Mr 

or 

.72 

\    \ 

1 

\\ 

V 

\\ 

\ 

\ 

\ 

\ 

A 

\v 

\ 

\ 

\ 

A 

\1 

L       \ 
\ 

\j 

^ 

\ 

DI= 

2;96 

\  N 

,  \ 

\ 

\ 

\ 

V 

\ 

\ 

\  ' 

\ 

s 

\ 

V 

i 

yi» 
'*- 

> 

•<5> 

y> 
'& 

\ 

\ 

B?r 

3i=] 

6.20 

r 

\ 

V 

\ 

\ 

V 

^c 

\ 

\ 

N 

^ 

\ 

\ 

—I 

onD 

i=19 

i4^ 

>l 

V 

vs 

Nn 

V 

% 

p? 

r 

170 


Railroad  Curves  and  Earthwork. 


293.   For  Use. 

Find  the  diagonal  line  corresponding  to  the  given  value  of 
CQ  —  Ci ;  follow  this  up  until  the  vertical  line  representing  the 
given  value  of  DQ  —  D\  is  reached,  and  the  intersection  is  thus 
found.  Then  read  off  the  value  of  C  corresponding  to  this 
intersection. 

Example.        CQ  —  c\  —    1.2 


again, 


c0  -  ei  =  1.7 
D0-Dl  =  7.0 


C  =  3.6  ± 


294.   Diagram  for  Triangular  Prisms. 

From  formula  (191),  8  =  — cD,  a  table  may  be  constructed. 


0 

5.4 

10.8 

16.2 

21.6 

27.0 

D 

0 

0 

0 

0 

0 

0 

0 

1 

0 

5 

10 

15 

20 

25 

2 

0 

10 

20 

30 

40 

50 

3 

0 

15 

30 

45 

60 

75 

4 

0 

20 

40 

60 

80 

100 

5 

0 

25 

50 

75 

100 

125 

6 

0 

30 

60 

90 

120 

150 

7 

0 

35 

70 

105 

140 

175 

8 

0 

40 

80 

120 

160 

200 

9 

0 

45 

90 

135 

180 

225 

10 

0 

50 

100 

150 

200 

250 

c 

From  this  a  diagram  can  be  constructed  similar  in  form  to 
that  for  Prismoidal  Correction. 

The  lines  for  all  values  of  c  pass  through  the  origin. 

In  constructing  this  table,  any  values  of  D  might  have  been 
taken  instead  of  those  used  here.  Those  used  were  selected 
because  they  give  results  simple  in  value,  easily  obtained,  and 
readily  platted. 


Earthwork  Diagrams.  171 

295.  Diagram  for  Three-Level  Sections. 

Formula,        *  =  £(e  +  ±)z>_»  .±  .  5  (192) 

A  separate  diagram  will  be  required  for  each  value  (or  com- 
bination of  values)  of  b  and  s.  Since  b  and  s  thus  become 
constants,  the  formula  assumes  the  form  of 

x  =  a(z  +  6)y  +  d  (197) 

and  the  diagram  will  consist  of  a  series  of  right  lines. 

A  table  can  be  made  up  by  taking  successive  values  of  c  =  0, 
1,  2,  3,  4,  etc.,  and  finding  for  each  of  these  the  value  of  S 
corresponding  to  different  values  of  D,  using  the  above  formula. 

To  make  separate  and  complete  computations  directly  by 
formula  would  be  quite  laborious  ;  there  is,  however,  a  method 
of  systematizing  the  construction  of  the  table  which  can  be 
shown  better  by  example  than  in  any  other  way. 

296.  Example.     6  =  14  s  =  1£  to  1 

Formula  S  =  ™'  c  +  ±D  -  §2  .  A  .  b 


54V        2s  54     2s 


becomes  8  =        c  +        D  _       .       .  14 

54V         3/          54     3 


S  =        e  +        #-  00.49  (198) 

54V          o  / 

A  table  has  been  prepared  for  successive  values  of 

c  =  0,        1,        2,        3,        4,        5,        etc. 
and  for        D  =  14,          16.2,         21.6,         27.0,         etc. 

These  values  of  Z>  are  selected  for  the  following  reasons  : 
D  =  14  is  the  least  possible  value  ;  D  =  16.2,  21.6  are  desirable 
because  they  are  multiples  of  5.4,  and  the  factors  in  the  for- 
mula show  that  the  computations  will  be  simplified  by  selecting 
multiples  of  5.4  for  the  successive  values  of  D. 


172 


Railroad  Curves  and  Earthwork. 


14 

16.2 

21.6 

27.0 

32.4 

37.8 

43.2 

D 

12.963 

15. 

20. 

25. 

30. 

35. 

40. 

Const, 
diff. 

0 

0 

9.51 

32.84 

56.18 

79.51 

102.84 

126.18 

1 

12.963 

24.51 

52.84 

81.18 

109.51 

137.84 

166.18 

2 

25.926 

39.51 

72.84 

106.18 

139.51 

172.84 

206.18 

3. 

38.889 

54.51 

92.84 

131.18 

169.51 

207.84 

246.18 

4 

51.852 

69.51 

112.84 

156.18 

199.51 

242.84 

286.18 

5 

64.815 

84.51 

132.84 

181.18 

229.51 

277.84 

326.18 

6 

77.778 

99.51 

152.84 

206.18 

259.51 

312.84 

366.18 

7 

90.741 

114.51 

172.84 

231.18 

289.51 

347.84 

406.18 

8 

103.704 

129.51 

192.84 

256.18 

319.51 

382.84 

446.18 

9 

116.667 

144.51 

212.84 

281.18 

349.51 

417.84 

486.18 

10 

129.630 

159.51 

232.84 

306.18 

379.51 

452.84 

526.18 

c 

When 
When 


c  =  0         8  =  &  -  ^  •  D  -  60.49 
=  U       tf  =  $£.#  .14-  60.49 

=     60.49      -  60.49  =  0 


When  D  =  16.2 

we  may  again  calculate  directly 

#  =  H-¥-16-2-  60-49 

but  a  better  method  is  to  find  how  much  greater  8  will  be  for 
D  =  16.2  than  for  D  =  14.0. 

We  have  s  =  $$-*£-D-  60.49 

Then  for  any  new  value  D' 

S'  =  $$.i£.D'-  60.49 

ff-8  =  M  .^(D'-D)  (199) 

for  IX  =  16.2        Z>  =  14.0        D'  -  D  =  2.2 

8'  -  8  =      •      x  2-2  =  9-51 


8'  =  9.51,  which  is  entered  in  table. 


Earthwork  Diagrams.  173 


Similarly,  S"  -  S'  =  ^  -  ^(D"  -  D') 

D"  =  21.6          D<  =  16.2          D"  -  D'  =  5.4 
S"  -  £'  =  f£  x  Y  x  5.4 

=  23.333 

S'=    9.51  fliv  =    79.509 

S"  =  32.843  23.333 

Similarly,          S'»  -  S"  =  23.333  £*  =  102.842 

5"'  =  56.176  23.333 

fl*  -  S"'  =  23.333  S*  =  126.175 

S"  =  79.509 
Constant  increment  forZ>'-  Z>  =  5.4  is  23.333. 

297.  Each  result  is  entered  in  the  table  in  its  proper  place. 
The  final  result  for  c  =  0  and  D  =  43.2  should  be  calculated 

independently  as  a  check. 

When          c  =  0  S=™.I*.D       -60.49 

When        Z)  =  43.2       8~  M  •  ¥  *  *3.2  -  60.49 

=  50  x  -1/  x  0.8  -  60.49 

=  56^  _  60-49 

I 

=  186.67  -  60.49 

S=  126.18 

This  checks  exactly,  and  all  intermediate  values  are  checked 
by  this  process,  which  is  also  more  rapid  than  an  independent 
calculation  for  each  value  of  D. 

298.  We  now  have  values  of  S  for  the  various  values  of 
D  =  14.0,   16.2,   21.6,  etc.,  when  c  =  0. 

Next,  find  how  much  these  will  be  increased  when  c  =  1. 

Formula  8  =  $$(c  +  *£)D  -  60.49 

for  any  new  value  c'         8'  -  ff  (c'  +  J^)D  -  60.49 

S'-fl=$$(c'-c)Z>  (200) 


174         Railroad  Curves  and  Earthwork. 

When         c'  =  1  and    c  =  0,  c'  -  c  =  1 

ff-8=#D 

Similarly,  8"  -  8'  =  if  (c"  -  c')Z> 

When         c"  =  2  and  c'  =  1,  c"  -  c'  =  1 

ff' ~  4*9  i|I> 

That  is,  for  any  increase  of  1  ft.  in  the  value  of  c, 

S'-S  =  &D  (201) 

When  D  =  14 

£'_S=5o  x  14  =  12.963 

This  we  enter  as  the  constant  difference  for  column  D  =  14. 

We  have  already  found  So  =   0 

12.963 

Si  =  12.963 

12.963 

£2  =  25.926 

This  gives  column  14.  #3  =  38.889  etc. 

When  D  =  16.2 

(201)  8'  -  S  =  f f  D  =  f£  x  16.2  =  50  x  0.3 

=  15 
Enter  15  as  the  constant  difference  in  column  16.2. 

We  already  have  /So  =   9.51 

15. 
&  =  24.51 

15. 
#2  =  39.51 

This  allows  us  to  complete  column  16.2.     83  —  54.51  etc. 

Similarly  for         D  =  21.6  8'  -  S  =  20 

Enter  20  as  constant  difference  in  column  21.6,  and  complete 
column  as  shown  in  table. 

Similarly,  fill  out  all  the  columns  shown  in  the  table. 


Earthwork  Diagrams.  175 

299.  The  final  result  for  c  =  10,  D  =  43.2  should  be  calcu- 
lated independently,  and  directly  from  the  formula,  as  a  check. 

£=f£(c+¥)#  -60.49 

c  =  10  D  =  43.2 

S=  fo  x  H.667  x  43.2  -  60.49 

=  50  x  14.667  x    0.8  -  60.49 

=  40  x  14.667  -  60.49 

=  586.68  -  60.49 

8=  526.19 

The  table  gives  526.18.  This  checks  sufficiently  close  to  indi- 
cate that  no  error  has  been  made.  It  would  yield  an  exact 
check  if  we  took  c  +  Y  =  14.6667. 

300.  Note  that  for      c  =  10        D  —  43.2        value  =  526.  18 

c  =  10        D  =  37.8  "         452.84 

Diff.    =    73.34 

Between  c  =  10 

and  c  £=  10 


Z>  =  37.81     Diff    =    73  33 
D  =  32.4  / 


In  line  c  =  10  a  constant  difference  is  found  between  succes- 
sive values  of  D  differing  by  5.4.  This  may  be  demonstrated 
to  be  =  73.33. 

All  values  in  the  table  except  column  14  are  satisfactorily 
checked  by  applying  this  difference  of  73.33  in  line  10  together 
with  the  independent  check  of  c  =  10,  D  =  43.2. 

The  value  of  c  =  10,  D  =  14  can  also  be  checked  and  shown 
to  be  correct. 

301.  Having  the  table,  page  168,  completed,  the  construction 
of  the  diagram  is  simple. 

The  "Diagram  for  Three-Level  Sections,  Base  14,  Slope  1^ 
to  V  was  calculated  and  constructed  according  to  this  table. 
The  Diagram  given  shows  a  general  arrangement  of  lines  and 
figures  convenient  for  use.  For  rapid  and  convenient  use,  the 
diagram  should  be  constructed  upon  cross-section  paper,  Plate 
G  ;  and  in  this  case  the  diagram  will  be  upon  a  scale  twice  that 
of  the  diagram  accompanying  these  notes. 


176         Railroad  Curves  and  Earthwork. 

A  "  curve  of  level  section  "  has  been  platted  on  this  diagram 
in  the  following  manner.  For  level  sections,  when 

c  =  0  D  -  14.0  c  =  2  D  =  20.0 

c  =  1  D  =  17.0  c  =  6  D  =  32.0 

.c  =  1.4         D  =  18.2  etc. 

The  line  passing  through  these  points  gives  the  "curve  of 
level  section." 

Aside  from  the  direct  use  of  this  curve  of  level  section  (for 
preliminary  estimates  or  otherwise),  it  is  very  useful  in  tending 
to  prevent  any  gross  errors  in  the  use  of  the  table,  since,  in 
general,  the  points  (intersections)  used  in  the  diagram  will  lie 
not  far  from  the  curve  of  level  section. 

302.   Use  of  Diagram. 

Find  the  diagonal  line  corresponding  to  the  given  value  of  c  ; 
follow  this  up  until  the  vertical  line  representing  the  given  value 
of  D  is  reached,  and  this  intersection  found.  Then  read  off  the 
value  of  8  corresponding  to  this  intersection. 

Example.    Notes. 


Sta.  I        -  -3.7  ; 


Sta.0  -2.5 


8  =214. 

For      Sta.  I     c  =  3.7  D  =  25.4 

c  =  3.7  is  the  middle  of  the  space  between  3.6  and  3.8. 
Follow  this  up  until  the  vertical  line  25.4  is  reached. 
The  intersection  lies  upon  the  line  Si  =  136. 
Enter  this  above  opposite  Sta,  1. 
For      Sta.  0      c  =  2.5    „        D  -  20.9 
c  =  2.5  is  the  middle  of  space  between  2.4  and  2.6. 
Follow  this  up  until  the  middle  of  space  between  20.8  and 
21.0  is  reached. 
The  intersection  lies  just  above  the  line 

#o=78 
Enter  this  opposite  Sta.  0. 

#100  =  #1    +  #0 

=  136  +  78  =  214  cu.  yds. 


Earthwork  Diagrams.  177 

The  prismoidal  correction  may  be  applied  if  desired. 
It  should  be  noticed  that  in  each  case  the  intersection  was 
quite  close  to  the  "  curve  of  level  section." 

303.  Diagrams  may  be  constructed  in  this  way  that  will 
give  results  to  a  greater  degree  of  precision  than  is  warranted 
by  the  precision  reached  in  taking  the  measurements  on  the 
ground. 

In  point  of  rapidity  diagrams  are  much  more  rapid  than  tables 
for  the  computation  of  Three-Level  Sections. 

For  "Triangular  Prisms"  and  for  Prismoidal  Correction, 
the  diagrams  are  somewhat  more  rapid. 

For  Level  Sections,  the  tables  for  Three  Level-Sections,  §  274, 
are  at  least  equally  rapid. 

A  book  entitled  "Computation  from  Diagrams  of  Railway 
Earthwork,"  by  Arthur  M.  Wellington,  published  by  D.  Apple- 
ton  &  Co.,  N.Y.,  explains  the  application  and  construction  of 
certain  other  tables  in  addition  to  those  given  here.  "  Welling- 
ton's Diagrams,"  as  there  published,  are  upon  a  scale  differing 
from  that  used  here,  and  they  do  not  allow  of  as  great  precision, 
but,  on  the  other  hand,  are  arranged  to  cover  a  large  number  of 
tables  differing  somewhat  as  to  base  and  slope. 

304.  The  use  of  approximate  methods  for  applying  the  pris- 
moidal correction  to  irregular  sections  (pp.  136-137)  will  be 
rendered  practicable  by  the  use  of  these  "  Diagrams  for  Three- 
Level  Sections." 

Method  1.     No  use  of  diagrams  is  necessary. 

Method  2.  Having  found  for  any  irregular  sections  (by  tri- 
angular prisms  or  any  other  method)  the  solidity  8  for  50  ft. 
in  length,  find  upon  the  diagram  the  line  corresponding  to 
this  value  of  S;  follow  this  line  to  tlxe  curve  of  level  section, 
and  read  off  the  value  of  c  (for  a  level  section)  which  corre- 
sponds, and  also  the  value  of  D  for  the  same  section. 

Method  3.  Having  found  in  any  way  the  value  of  S ;  if  c  is 
given,  find  the  value  of  D  to  correspond  ;  if  D  is  given,  find  the 
value  of  c  to  correspond. 

Method  4.     The  use  of  diagrams  is  not  needed. 


CHAPTER  XVI. 
HAUL. 

305.  When  material  from  excavation  is  hauled  to  be  placed 
in  embankment,  it  is  customary  to  pay  to  the  contractor  a 
certain  sum  for  every  cubic  yard  hauled.     Oftentimes  it  is  pro- 
vided that  no  payment  shall  be  made  for  material  hauled  less 
than  a  specified  distance.     In  the  east  a  common  limit  of  "free 
haul"  is  1000  ft.      Often  in  the  west  100  ft.  is  the  limit  of 
"free  haul." 

A  common  custom  is  to  make  the  unit  for  payment  of  haul, 
one  yard  hauled  100  ft.  ;  the  price  paid  will  often  be  from  1  to 
2  cents  per  cubic  yard  hauled  100  ft. 

The  price  paid  for  "haul "  is  small,  and  therefore  the  stand- 
ard of  precision  in  calculation  need  not  be  quite  as  fine  as  in 
the  calculation  of  the  quantities  of  earthwork.  The  total 
"  haul "  will  be  the  product  of 

(1)  the  total  amount  of  excavation  hauled,  and 

(2)  the  average  length  of  haul. 

306.  The  average  length  of  haul  is  the  distance  between  the 
center  of  gravity  of  the  material  as  found  in  excavation,  and 
the  center  of  gravity  as  deposited.     It  would  not,  in  general,  be 
simple  to  find  the  center  of  gravity  of  the  entire  mass  of  exca- 
vation hauled,  and  the  most  convenient  way  is  to  take  each 
section  of  earthwork  by  itself.    The  "haul"  for  each  section 
is  the  product  of  the 

(1)  number  of  cubic  yards  in  that  section,  and 

(2)  distance  between  the  center  of  gravity  in  excavation, 

and  the  center  of  gravity  as  deposited. 
178 


Haul. 


179 


307.  When  excavation  is  placed  in  embankment,  there  may 
be  some  difficulty  in  determining  just  where  any  given  section 
of  excavation  will  be  placed,  and  where  its  center  of  gravity 
will  be  in  embankment. 


B 

In  hauling  excavation  in  embankment,  there  is  some  plane, 
as  indicated  by  AB,  to  which  all  excavation  must  be  hauled  on 
its  way  to  be  placed  in  embankment,  and  (another  way  of  put- 
ting it)  from  which  all  material  placed  in  embankment  must 
be  hauled  on  its  way  from  excavation.  We  may  figure  the 
total  haul  as  the  sum  of 

(1)  total  "haul"  of  excavation  to  AB,  and 

(2)  total  "haul"  of  embankment  from  AB. 

The  total  "haul"  of  excavation  to  AB  and  the  total  "haul " 
of  embankment  from  AB  will  most  conveniently  be  calculated 
as  the  sum  of  the  hauls  of  the  several  sections  of  earthwork. 
Tor  each  section  the  haul  is  the  product  of 

(1)  the  solidity  8  of  that  section,  and 

(2)  distance  from  center  of  gravity  of  that  section  to  the 

plane  AB. 

308.  When  the  two  end  areas  are  equal,  the  center  of  gravity 
will  be  midway  between  the  two  end  planes.  When  the  two  end 
areas  are  not  equal  in  value,  the  center  of  gravity  of  the  section 
will  be  at  a  certain  distance  from  the  mid-section,  as  shown 
by  the  formula 

V     A^-Ay 
g~'    ~~ 


in  which  xg  =  distance  center  of  gravity  from  mid-section. 


180         Railroad  Curves  and  Earthwork. 


309.  Referring  to  the  figure  below,  and  following  the  same 
general  method  of  demonstration  used  on  page  230,  §  235, 

let  60  =      base      =  A0B0 

c0  =  center  ht.  =  00G0 
Ci  =  center  ht.=OiGi 
I  =  length  (altitude) 
of  section  =  GoGi 
AQ  =  area  of  DoA0B0Eo 
AI  —  area  of  DiAiBiEi 
.&•=.  solidity 

"0  «0  B0 

Also  use  notation  bx,  cz,  Ax  for  a  section  distant  x  from  GI. 

Find  the  distance  of  the  center  of  gravity  from  AiBiEiDi,  and 
let  this  =  xc.  Let  xg  =  distance  of  center  of  gravity  from  mid- 
section. 

Then  for  any  elementary  section  of  thickness  dx  and  distance 
x  from  AiBiEiDi,  its  moment  will  be 


+  (Co  -  ci)      x  dx 


+  (60  - 


+  (60  -  61)  *      ci  +  (c0  -  ci)       x  dx 


2  '     3! 

3  I          4  Z* 

6  &iCi  +  4  &iCo  +  4 

&oCi  -f  3  &0Co 

Z2 

-  4  biCi  -  3  6iC0  -  3 

&OC1 

12 

—  4  b\c\ 

+  3  bici 

-   x 

I2    .. 
X 


6pCi  +  3  60c0 


S 


(202) 


Haul  181 

What  is  wanted  is  xg  rather  than  «c. 

Xg=  l~~         Xe 


=S--  Sxe 
2 


S  =  -(2  &ici  +  2  &0co  +  Mo  +  &oCi)          (164) 
o 

I  =  -£-(2  &ici  +  2  &0co  +  &ic0  +  &oCi)          (203) 


8  .  zc  =    (6ici  +  3  &0co  +  &ico  -f  &od) 


'(Sin  cu.  ft.)  (204) 

(205) 

310.  The  formula  above  applies  to  the  solid  shown  in  the 
figure,  which  has  trapezoidal  ends,  but  it  will  apply  also  when 
D0Ao,  DiAi,  are  each  =  0,  and  therefore  applies  to  such  solids 
having  triangular  ends ;  and  since  any  section  of  earthwork 
with  parallel  ends  may  be  divided  into  a  number  of  such  solids 
with  triangular  ends,  it  applies  to  all  ordinary  sections  of  rail- 
road earthwork,  since  it  applies  to  the  parts  of  which  it  is 
made  up. 

To  show  that  in  fact  this  formula  is  correct  for  prisms,  wedges, 
and  pyramids,  use  a  method  similar  to  that  shown  on  page  129  ; 
find  for  each  solid  an  expression  for  xg  in  terms  of  A  and  I ; 
reduce  to  the  form 

f.*-4. 
12          S 


182         Railroad  Curves  and  Earthwork. 

311.    The  formula 

V         A,  -A, 
9      12  x  27  '        8 

is  not  in  form  convenient  for  use,  because  we  have  not  found 
the  values  of  AI  and  AQ,  but  instead  have  calculated  directly 
from  the  tables  or  diagrams,  the  values  of  Si  and  SQ  for  50  ft. 
in  length,  where 

50  27  Si 


and  (          A.  = 

substituting,  .i 


This  formula  is  in  shape  convenient  for  use,  and  results 
correct  to  the  nearest  foot  can  be  calculated  with  rapidity. 

312.    For  a  section  of  length  Z  less  than  100  ft. 


12x27  St 

p  Ai  -  AQ 


*•"» 

IPO  i       AI  -A0 

12  x  27 


100  x  100      AI-  A 


ff™        12  x  27  #100 

srtL.'-L  (207) 


Haul. 


183 


313.  It  is  not,  however,  always  necessary  to  calculate  the 
position  of  the  center  of  gravity  of  each  station,  or  to  calculate 
for  each  station  the  correction  xg.  It  may  often  be  easy  to 
calculate  for  a  series  of  sections  a  correction  to  be  applied 
to  obtain  the  center  of  gravity  of  the  entire  mass. 


•<     100    > 


To  find  the  position  of  the  center  of  gravity  of  the  entire 
mass,  let 

JTC  =  cent,  of  grav.  for  entire  mass  (approximately), 
using  for  each  section  e.g.  at  - 

X  =  true  dist.  to  e.g.  of  entire  mass 
Xg=X-Xc 

So  =  ff  AQ,  Si  =  ^  AI,  S2  =  etc.,  as  taken  from  tables 
or  diagrams. 

When  all  sections  are  of  uniform  length   =  I  as  in  figure 
above, 


ScXg 


SXff  =  ^  ($>-£ 


100 


or,  in  general, 


(208) 


where  S  is  the  solidity  of  the  entire  mass. 


CHAPTER  XVII. 
MASS  DIAGRAM. 

314.  Many  questions  of  "  haul "  may  be  very  usefully  treated 
by  means  of  a  graphical  method,  known  to  some  as  "Mass 
Leveling,"  in  which  is  used  a  diagram  sometimes  called  a 
"Mass  Profile,"  but  which  will  be  referred  to  here  as  the 
"Mass  Diagram." 

The  construction  of  the  "  Mass  Diagram  "  will  be  more  clearly 
understood  from  an  example  than  from  a  general  description. 

Let  us  consider  the  earthwork  shown  by  the  profile  on  p.  182, 
consisting  of  alternate  "cut"  and  "fill."  To  show  the  work 
of  constructing  the  ' '  diagram ' '  in  full,  the  quantities  are  calcu- 
lated throughout,  but  for  convenience  and  simplicity,  "level 
sections"  are  used  and  prismoidal  correction  disregarded.  In 
a  case  in  actual  practice,  the  solidities  will  have  been  calculated 
for  the  actual  notes  taken. 

315.  In  the  table,  p.  181,  the 
1st  column  gives  the  station. 
2d  column  gives  center  heights. 

3d  column  gives  values  of  S  from  tables. 

4th  column  gives  values  of  $100  or  Si  for  each  section,  and 
with  sign  +  for  cut  or  —  for  fill. 

5th  column  gives  the  total,  or  the  sum  of  solidities  up  to  each 
station  ;  and  in  getting  this  total,  each  +  solidity  is  added  and 
each  —  solidity  is  subtracted,  as  appears  in  the  table  from  the 
results  obtained. 

Having  completed  the  table,  the  next  step  is  the  construction 
of  the  "  Mass  Diagram,"  page  182.  In  the  figure  shown  there, 
each  station  line  is  projected  down,  and  the  value  from  column 
5,  corresponding  to  each  station,  is  platted  to  scale  as  an  offset 
from  the  base  line  at  that  station,  all  +  quantities  above  the 
line,  and  all  —  quantities  below  the  line.  The  points  thus  found 
are  joined,  and  the  result  is  the  "  Mass  Diagram." 

184 


Mass  Diagram. 


185 


STATION. 

CENTER 
HEIGHTS. 

SOLIDITY  FOB 

50'  DUE  TO 

CENTER 
HEIGHT  GIVEN 
(TAKEN  FROM 
TABLES). 

SOLIDITY 

FOE 

SECTION. 

SOLIDITY 
TOTALS. 

0 

0 

0 

0 

1 

+  1.7 

71 

+  71 

+  71 

2 

+  2.7 

120 

+  191 

+  262 

3 

0 

0 

+  120 

+  382 

4 

-3.3 

116 

-116 

+  266 

5 

-5.1 

204 

-320 

-  54 

6 

-2.9 

99 

-303 

-  357 

7 

0 

0 

-  99 

-  456 

8 

+  2.4 

105 

+  105 

-  351 

9 

+  4.5 

223 

+  328 

-  23 

10 

+  2.5 

110 

+  333 

+  310 

11 

0 

0 

+  110 

+  420  , 

12 

-3.0 

.  103 

-103 

+  317 

13 

-5.3 

215 

-318 

-   1 

14 

-7.6 

357 

-572 

-  573 

15 

-8.4 

414 

-771 

-1344 

16 

-4.3 

163 

-577 

-1921 

17 

0 

0 

-163 

-2084 

18 

+  2.6 

115 

+  115 

-1969 

19 

+  3.6 

169 

+  284 

-1685 

20 

+  4.9 

248 

+  417 

-1268 

21 

+  6.7 

373 

+  621 

-  647 

22 

+  7.5 

434 

+  807 

+  160 

23 

+  5.2 

268 

+  702 

+  862 

24 

+  2.4 

105 

+  373 

+  1235 

25 

0 

0 

+  105 

+  1340 

26 

-3.6 

129 

-129 

+  1211 

27 

-6.0 

256 

-385 

+  826 

28 

-5.0 

199 

-455 

+  371 

29 

-2.6 

86 

-285 

+  86 

30 

0 

0 

-  86 

0 

186         Railroad   Curves  and  Earthwork. 


\ 


1021 


2Q& 
1969 
1685 


54          £ 


Mass  Diagram. 


187 


316.  It  will  follow,  from  the  methods  of  calculation  and  con- 
struction used,  that  the  "  Mass  Diagram"  will  have  the  follow- 
ing properties,  which  can  he  understood  by  reference  to  the 
profile  and  diagram,  page  182. 

1.  Grade  points  of  the  profile  correspond  to  maximum  and 
minimum  points  of  the  diagram. 

2.  In  the  diagram,  ascending  lines  mark  excavation,  and  de- 
scending lines  embankment. 

3.  The  difference  in  length  between  any  two  vertical  ordinates 
of  the  diagram  is  the  solidity  between  the  points  (stations)  at 
which  the  ordinates  are  erected. 

4.  Between  any  two  points  where  the  diagram  is  intersected 
by  any  horizontal  line,  excavation  equals  embankment. 

5.  The  area  cut  off  by  any  horizontal  line  is  the  measure  of 
the  "  haul "  between  the  two  points  cut  by  that  line. 

317.  It  may  be  necessary  to  explain  the  latter  point  at  some- 
what greater  length. 

Any  quantity  (such  as  dimension,  weight,  or  volume)  is  often 
represented  graphically  by  a  line ;  in  a  similar  way,  the  product 
of  two  quantities  (such  as  volume  into  distance,  or  as  foot 
pounds)  may  be  represented  or  measured  by  an  area.  In  the 
case  of  a  figure  other  than  a  rectangle,  the  value,  or  product 
measured  by  this  area,  may  be  found  by  cutting  up  the  area  by 
lines,  and  these  lines  may  be  vertical  lines  representing  volumes 
or  horizontal  lines  representing  distance.  The  result  will  be 
the  same  in  either  case.  An  example  will  illustrate. 

In  the  two  figures  let 
a  and  b  represent  pounds 
c  "         feet 

and  the  area  of  the  trapezoid 
represent  a  certain  number 
c  c  of  foot  pounds.     The  trape- 

zoid may  be  resolved  into  rectangles  by  the  use  of  a  vertical 
line,  as  shown  in  Fig.  1,  or  by  a  horizontal  line,  as  in  Fig.  2. 


In  Fig.  1,  the  area  is 
In  Fig.  2,  the  area  is 


x  c 


the  result  of  course  being  the  same  in  both  cases. 


188         Railroad  Curves  and  Earthwork. 


Mass  Diagram.  189 

318.  In  an  entirely  similar  way,  the  area  ABC  (p.  184)  repre- 
sents the  "haul"  of  earthwork  (in  cu.  yds.  moved  100  ft.) 
between  A  and  C,  and  this  area  may  be  calculated  by  dividing 
it  by  a  series  of  vertical  lines  representing  solidities,  as  shown 
above  G  and  F.     That  this  area  represents  the  haul  between 
A  and  C  may  be  shown  as  follows  :  — 

Take  any  elementary  solidity  dS  at  D.  Project  this  down 
upon* the  diagram  at  F,  and  draw  the  horizontal  lines  FG. 

Between  the  points  F  and  G  (or  between  D  and  I),  there- 
fore, excavation  equals  embankment,  and  the  mass  dS  must  be 
hauled  a  distance  FG,  and  the  amount  of  "haul"  on  dS  will 
be  dS  x  FG,  measured  by  the  trapezoid  FG.  Similarly  with  any 
other  elementary  dS. 

The  total  "haul"  between  A  and  C  will  be  measured  by  the 
sum  of  the  series  of  trapezoids,  or  by  the  area  ABC.  This  area 
is  probably  most  conveniently  measured  by  the  trapezoids 
formed  by  the  vertical  lines  representing  solidities.  The  aver- 
age length  of  haul  will  be  this  area  divided  by  the  total  solidity 
(represented  in  this  case  on  p.  182  by  the  longest  vertical  line, 
2084). 

319.  The  construction  of  the  "Mass  Diagram"  as  a  series 
of  trapezoids  involves  the  assumption  that  the  center  of  gravity 
of  a  section  of  earthwork  lies  at  its  mid-section,  which  is  only 
approximately  correct  since  S  for  the  first  50  ft.  will  seldom  be 
exactly  the  same  as  S  for  the  second  50  ft.  of  a  section  100  ft. 
long.     If  the  lines  joining  the  ends  of  the  vertical  lines  be  made 
a  curved  line,  the  assumption  becomes  more  closely  accurate, 
and  if  the  area  be  calculated  by   "Simpson's  Kule,"  or  by 
planimeter,  results  closely  accurate  will  be  reached. 

It  will  be  further  noticed  that  hill  sections  in  the  "diagram  " 
represent  haul  forward  on  the  profile,  and  valley  sections  haul 
backward.  The  mass  diagram  may  therefore  be  used  to  indi- 
cate the  methods  by  which  the  work  shall  be  performed ; 
whether  excavation  at  any  point  shall  be  hauled  forward  or 
backward ;  and,  more  particularly,  to  show  the  point  where 
backward  haul  shall  cease  and  forward  haul  begin,  as  indicated 
in  the  figure,  p.  184,  which  shows  a  very  simple  case,  the  cuts 
and  fills  being  evenly  balanced,  and  no  haul  over  900  feet,  with 
no  necessity  for  either  borrowing  or  wasting. 


190          Railroad   Curves  and  Earthwork. 


Mass  Diagram.  191 

320.  In  the  figure,  page  186,  the  excavation  from  Sta.  0  to  14  is 
very  much  in  excess  of  embankment,  and  vice  versa  from  Sta.  14 
to  30.     The  mass  diagram  indicates  a  haul  of  nearly  3000  ft. 
for  a  large  mass  of  earthwork,  measured  by  the  ordinate  AB. 
It  will  not  be  economical  to  haul  the  material  3000  ft. ;  it  is 
better  to  "waste"  some  of  the  material  near  Sta.  0,  and  to 
"  borrow  "  some  near  Sta.  30,  if  this  be  possible,  as  is  commonly 
the  case. 

If  we  draw  the  line  CD,  the  cut  and  fill  between  C  and  D  will 
still  be  equal,  and  the  volume  of  cut  measured  by  CE  can  be 
wasted,  and  the  equal  volume  of  fill  measured  by  DF  can  be 
borrowed  to  advantage.  It  can  be  seen  that  there  is  still  a  haul 
of  nearly  2000  ft.  (from  A  to  D)  on  the  large  mass  of  earthwork 
measured  by  GH.  It  is  probable  that  it  will  not  pay  to  haul 
the  mass  GH,  or  any  part  of  it,  as  far  as  AD. 

321.  We  must  find  the  limit  beyond  which  it  is  unprofitable 
to  haul  material  rather  than  borrow  and  waste. 

Let    c  =  cost  of  1  cu.  yd.  excavation  or  embankment. 
h  =  cost  of  haul  on  1  cu.  yd.  hauled  100  ft. 
n  =  length  of  haul  in  "  stations"  of  100  ft.  each. 

Then,  when  1  cu.  yd.  of  excavation  is  wasted,  and  1  cu.  yd. 
of  embankment  is  borrowed, 

cost  =  2  c 
When  1  cu.  yd.  of  excavation  is  hauled  into  embankment, 

cost  =  c  -f  nh 
The  limit  of  profitable  haul  is  reached  when 

2  c  =  c  +  nh 

or  when  n=-  (209) 

h 

Example.  When  excavation  or  embankment  is  18  cents  per 
cu.  yd.,  and  haul  is  1|  cents, 

n  =  —  =  12  stations 
1.5 

When  c  =  16  and  h  =  2 

then  n  =  8  stations 


192         Railroad  Curves  and  Earthwork. 


Mass  Diagram.  193 

322.  In  the  former  case  (1200  ft.  haul)  we  should  draw  in 
mass  diagram  (p.  188)  the  line  KGL.     Here  KG  is  less  than 
1200  ft.    The  line  should  not  be  lower  than  G,  for  in  that  case 
the  haul  would  be  nearly  as  great  as  KL,  or  more  than  1200  ft. 

In  the  latter  case  (800  ft.  haul)  the  line  would  be  carried  up 
to  a  point  where  NM  =  800  ft.  The  masses  between  N  and  A, 
also  C  and  0,  can  better  be  wasted  than  hauled,  and  the  masses 
between  M  and  G,  also  L  and  Z,  can  better  be  borrowed  than 
hauled  (always  provided  that  there  are  suitable  places  at  hand 
for  borrowing  and  wasting) . 

Next,  produce  NM  to  R.  The  number  of  yards  borrowed 
will  be  the  same  whether  taken  at  RZ  or  at  MG  +  LZ.  That 
arrangement  of  work  which  gives  the  smallest  "haul"  (product 
of  cu.  yds.  x  distance  hauled)  is  the  best  arrangement.  The 
"haul  "  in  one  case  is  measured  by  GLRYU,  and  in  the  other 
by  MGU  +  UYR.  If  MGU  is  less  than  GLRU,  then  it  is  cheaper 
to  borrow  (a)  RZ  rather  than  (6)  MG  +  LZ. 

In  a  similar  way  material  NT  and  SO  can  be  wasted  more 
economically  than  NA  and  CO. 

The  most  economical  position  for  the  line  M  R  is  when  M  U  =  UR. 
For  ST,  when  SV  =  VT.  Jor  any  change  from  these  positions  of 
MR  and  ST  will  show  an  increase  of  area  representing  "haul." 

323.  The  case  is  often  not  as  simple  as  that  here  given. 
Very  often  the  material  borrowed  or  wasted  has  to  be  hauled 
beyond  the  limit  of  "  free  haul."    The  limit  beyond  which  it  is 
unprofitable  to  haul  will  vary  according  to  the  length  of  haul 
on  the  borrowed  or  wasted  material ;  the  limit  will,  in  general, 
be  increased  by  the  length  of  haul  on  the  borrowed  or  wasted 
material.     The  haul  on  wasted  or  borrowed  material,  as  NT, 
may  be  shown  graphically  by  NTXW,  where  NW  =  TX  shows 
the  length  of  haul,  and  NTXW  the  "haul "  (mass  x  distance). 

The  mass  diagram  can  be  used  also  for  finding  the  limit  of 
"free  haul"  on  the  profile,  and  various  applications  will  sug- 
gest themselves  to  those  who  become  familiar  with  its  use  and 
the  principles  of  its  construction.  Certainly  one  of  its  most 
important  uses  is  in  allowing  "haul "  and  " borrow  and  waste " 
to  be  studied  by  a  diagram  giving  a  comprehensive  view  of  the 
whole  situation.  There  are  few  if  any  other  available  methods 
of  accomplishing  this  result. 


194         Railroad  Curves  and  Earthwork. 


Table  for  Three-Level  Sections.    Base  14',  Slope  1|  to  1. 


0 

.1 

.2 

.3 

.4 

L 

K 

L 

AT 

L 

K 

L 

K 

L 

K 

o 

0. 

6.5 

2.6 

6.6 

5-3 

6.8 

8.0 

6.9 

10.8 

7.0 

O 

I 

28.7 

7-9 

3V9 

8.0 

35-i 

8.1 

38-4 

8.3 

41.7 

8.4 

I 

2 

63.0 

9-3 

66.7 

9-4 

70-5 

9-5 

74-3 

9-7 

78.2 

9.8 

2 

3 

102.8 

10.6 

107.1 

10.8 

111.4 

10.9 

115-8 

ii.  i 

120.3 

II.  2 

3 

4 

148.1 

12.0 

153.0 

12.2 

157-9 

12.3 

162.8 

12.5 

167.9 

12.6 

4 

5 

199.1 

13.4 

204.5 

13.6 

209.9 

J3-7 

215-4 

13-8 

221.0 

14.0 

5 

6 

255-6 

14.8 

261.5 

I5.0 

267.5 

!5  ! 

273.6 

15.2 

279.7 

15-4 

6 

7 

317-6 

16.2 

324.1 

l6.3 

330.7 

16.5 

337-3 

16.6 

344-0 

16.8 

7 

8 

385.2 

17.6 

392.2 

17.7 

399-4 

17.9 

406.5 

18.0 

413-8 

18.  i 

8 

9 

458.3 

19.0 

466.0 

I9.I 

473-6 

19-3 

481.4 

19.4 

489.1 

19-5 

9 

IO 

537-0 

20.4 

545-2 

20.5 

553-4 

20.  6 

561-7 

20.8 

570-1 

20.9 

10 

ii 

621.3 

21.8 

630.0 

21.9 

638.8 

22.0 

647.7 

22.2 

656.6 

22.3 

IT 

12 

711.1 

23.1 

720.4 

23-3 

729.7 

23-4 

739-  ! 

23-6 

748.6 

23-7 

1  1 

13 

806.5 

24.5 

816.3 

24.7 

826.2 

24.8 

836.2 

25-0 

846.2 

25-1 

i-.' 

14 

907.4 

25.9 

917.8 

26.1 

928.3 

26.2 

938.8 

26.3 

•  949-3 

26.5 

^tt 

15 

1013.9 

27-3 

1024.8 

27-5 

1035-9 

27.6 

1046.9 

27-7 

1058.0 

27.9 

i'5 

16 

1125.9 

28.7 

"37-4 

28.8 

1149.0 

29.0 

1160.6 

29.1 

1172.3 

29-3 

16 

17 

1243-5 

30.1 

1255-6 

30.2 

1267.7 

3°-4 

1279.9 

30-5 

1292.1 

30.6 

17 

18 

1366.7 

31.5 

1379.3 

31-6 

1392.0 

31-8 

1404.7 

I417-5 

32.0 

18 

19 

1495.4 

32.9 

1508.5 

33'0 

1521.8 

I535-I 

33-3 

1548.4 

33-4 

19 

20 

1629.6 

34-3 

1643.4 

34-4 

1657.1  34.5 

1671.0 

34-7 

1684.9 

34-8 

20 

O 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

L 

K 

L 

K 

L 

K 

L 

K 

L 

K 

0 

J3-7 

7-2 

16.6 

7-3 

19-5 

7-5 

22.5 

7-6 

25.6 

7-7 

0 

I 

45-i 

8.6 

48.6 

8-7 

52.1 

8.8 

55-7 

9.0 

59-3 

9.1 

i 

2 

82.2 

IO.O 

86.2 

IO.  I 

90.2 

10.2 

94-4 

10.4 

98.5  10.5 

2 

3 

124.8 

"•3 

129.3 

11.5 

134.0 

n.6 

138.6 

n.8 

143-4 

11.9 

3 

4 

172.9 

12.7 

178.0 

12.9 

183.2 

13.0 

188.4 

13.1 

193-7 

4 

5 

226.6 

14.1 

232.3 

14-3 

238.0 

14.4 

243.8 

i4-5 

249.7 

14.7 

5 

6 

285.9 

15-5 

292.1 

15.6 

298.4 

15-8 

304-7 

15.9 

311.1 

16.1 

6 

I 

350.7 
421.1 

16.9 
18.3 

357-5 
428.4 

17.0 
18.4 

364-3 
435-8 

17.2 
18.6 

371-2 
443-3 

ll\7 

378.2 
450.8 

is'.i 

I 

9 

497-o 

19.7 

504-9 

19.8 

512.8 

20.0 

520.9 

20.1 

528.9 

2O.  2 

9 

IO 

578.5 

21.  1 

586.9 

21.2 

595-4 

21-3 

604.0 

21-5 

612.6 

21.6 

IO 

ii 

665.5 

22.5 

674.5 

22.6 

683.6 

22.7 

692.7 

22.9 

701.9 

23-0 

ii 

12 

758.1 

23.8 

767.7 

24.0 

777-3 

24.1 

787.0 

24.3 

796.7  24.4 

12 

13 

856.2 

25.2 

866.4 

25-4 

876.5 

25-5 

886.8 

25.6 

897.1  |  25.8 

13 

14 

960.0 

26.6 

970.6 

26.8 

981.4 

26.9 

992.1 

27.0 

IOO3.O   27.2 

14 

15 

1069.2 

28.0 

1080.4 

28.1 

1091.7 

28.3 

1103.1 

28.4 

III4.5 

28.6 

15 

16 

1184.0 

29.4 

1195.8 

29-5 

1207.7 

29.7 

1219.6 

29.8 

1231.5 

3O.O 

16 

17 

I3°4-4 

30.8 

1316.7 

30.9 

1329.1 

31-1 

1341.6 

31.2 

I354-I 

31-3 

J7 

18 

1430.3 

32.2 

I443.2 

32-3 

1456.2 

32.5 

1469.2 

32.6 

1482.2 

18 

19 

1561.8 

33-6 

I575>3 

33-7 

1588.8 

33-8 

1602.3 

34-0 

1616.0 

34-i 

19 

20 

1693-8 

35-0 

1712.9 

35-i 

1726.9 

35-2 

1741.0 

35-4 

I755-2 

35-5 

20 

.5 

.6 

.7 

.8 

.9 

Tables. 


195 


Table  for  Three-Level  Sections.     Base  20',  Slope  1|  to  1. 


0 

.1 

.2 

.3 

.4 

L 

K 

L 

K 

L 

K 

L 

K 

L 

K 

o 

o. 

9-3 

3-7 

9.4 

7-5 

9-5 

11.4 

9-7 

iS-3 

9.8 

o 

I 

39-8 

10.6 

44.1 

10.8 

48.4 

10.9- 

52-8 

ii.  i 

57-3 

II.  2 

I 

2 

85.2 

12.  0 

90.0 

12.2 

94-9 

12.3 

99-9 

12.5 

104.9 

12.6 

2 

3 

136.1 

13.4 

I4I-5 

13-6 

147.0 

13-7 

152-5 

13-8 

158.0 

14.0 

3 

4 

192.6 

I4.8 

198.5 

15.0 

204.6 

15-1 

2IO.6 

15.2 

216.7 

x5-4 

4 

5 

254-6 

16.2 

261.1 

I6.3 

267.7 

16.5 

274-3 

16.6 

281.0 

16.8 

5 

6 

322.2 

17.6 

329-3 

17.7 

336.4 

17.9 

343-6 

18.0 

350.8 

18.1 

6 

7 

395-4 

19.0 

403-0 

I9.I 

410.7 

19-3 

4Io1 

19.4 

426.2 

19.5 

7 

8 

474.1 

20.4 

482.2 

20-5 

49°-5 

20.  6 

498.8 

20.8 

5Q7-1 

20.9 

8 

9 

558-3 

21.8 

567-1 

21.9 

22.0 

584-7 

22.2 

593-6 

22.3 

9 

10 

648.1 

23.1 

657-4 

23-3 

666.8 

23-4 

676.2 

23.6 

685.6 

23-7 

10 

743-5 

24-5 

753-4 

24.7 

763-3 

24.8 

773-2 

25.0 

783.2 

25.1 

ii 

*! 

844.4 
950-9 

25-9 
27.3 

854.8 
961.9 

26.1 
27-5 

865-3 
972.9 

26.2 
27.6 

875.8 
984.0 

26.3 
27-7 

886.4 
995-i 

26.5 
27.9 

12 

13 

.  ,  1063.0 

28.7 

i074.5 

28.8 

1086.0 

29.0 

1097.7 

29.1 

1109.3 

29-3 

H 

15 

1180.6 

30.1 

1192.6 

30.2 

1204.7 

3°-4 

1216.9 

30.5 

1229.1 

30.6 

15 

16 

1303-7 

31.5 

1316.3 

31.6 

1329.0 

31.8 

i34i-7 

31-9 

1354-5 

32.0 

16 

17 

1432.4 

32-9 

1445.6 

33-o 

1458.8 

33-1 

1472.1 

33-3 

1485.4 

33-4 

17 

18 

1566.7 

34-3 

1580.4 

34-4 

1594.2 

34-5 

1608.0 

34-7 

1621.9 

34-8 

18 

19 

1706.5 

35-6 

1720.8 

35-8 

I735-1 

35-9 

*749-5 

36.1 

1764.0 

36.2 

19 

20 

1851.9 

37-o 

1866.7 

37-2 

1881.6 

37-3 

1896.5 

37-5 

1911.6 

37-6 

20 

O 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

L 

K 

L 

K 

L 

K 

L 

K 

L 

K 

O 

19.2 

10.0 

23.2 

10.  1 

27-3 

10.2 

31-4 

10.4 

35-6 

10.5 

o 

I 

61.8 

"•3 

66.4 

"•5 

71.0 

n.6 

75-7 

xi.8 

80.4 

11.9 

i 

2 

IIO.O 

12.7 

115.1 

12.9 

120.2 

13.0 

125.5 

13-1 

130-8 

i3-3 

2 

3 

l63-7 

14.1 

169.3 

T4-3 

I75-i 

14.4 

180.9 

14-5 

186.7 

14.7 

3 

4 

222.9 

15-5 

229.1 

15-6 

235-4 

15.8 

241.8 

i5-9 

248.2 

16.1 

4 

5 

287.7 

16.9 

294-5 

17.0 

301.4 

17.2 

308.3 

i7-3 

315-2 

17-5 

5 

6 

358.r 

18.3 

365^4 

18.4 

372.8 

18.6 

380.3 

18.7 

387-8 

18.8 

6 

7 

434-0 

19.7 

441.9 

19.8 

449.9 

2O.O 

457-9 

20.  i 

466.0 

20.2 

7 

8 

5I5-5 

21.  1 

524.0 

21.2 

532.5 

21-3 

541.0 

21-5 

549-7 

21.6 

8 

9 

602.5 

22-5 

611.6 

22.6 

620.6 

22.7 

629.7 

22.9 

638.9 

23.0 

9 

10 

695.1 

23-8 

704.7 

24.0 

714.3 

24.1 

724.0 

24-3 

733-7 

24.4 

10 

ii 

12 

793-3 
897.0 

25.2 

26.6 

803.4 
907.7 

25-4 

26.8 

813.6 
918.4 

25-5 
26.9 

823.8 
929.2 

25.6 
27.0 

834.1 
940.0 

25-8 
27.2 

ii 

12 

13 

1006.2 

28.0 

1017.5 

28.1 

1028.8 

28.3 

1040.1 

28.4 

°5I-5 

28.6 

13 

14 

II2I.I 

29-4 

1132.9 

29-5 

1144.7 

29.7 

1156.6 

29.8 

168.5 

30.0 

M 

15 

1241.4 

30.8 

1253.8 

30.9 

1266.2 

31-1 

1278.6 

31.2 

291.1 

31-3 

15 

16 

I367-4 

32.2 

1380.3 

32.3 

1393-2 

32.5 

1406.2 

32.6 

4I9-3 

32-7 

16 

17 

1498.8 

33-6 

1512.3 

33-7 

1525-8 

33-8 

J539-4 

34-o 

553-0 

34-  1 

17 

18 
19 

1635.9 
1778.5 

35-0 
36.3 

1649.9 
1793-0 

35-i 
36.5 

1664.0 
1807.7 

US 

1678.1 
1822.3 

m 

692.2 
837-I 

35-5 
36.9 

18 
19 

20 

1926.6 

37-7 

1941.7 

37-9 

1956.9 

38.0 

1972.1 

38.1 

1987.4 

38.3 

20 

.5 

.6 

.7 

.8 

.9 

196 


Railroad   Curves  and  Earthwork. 


Table  of  Prismoidal  Corrections. 


c,-c, 

1 

2 

3 

4 

5 

6 

7 

8 

9 

C.-C, 

A-A 

O.I 

•03 

.06 

.09 

.12 

•15 

.19 

.22 

•25 

.28 

O.I 

.2 

.06 

.12 

.19 

•25 

•37 

•43 

•49 

•56 

.2 

•3 

.09 

.19 

.28 

•37 

'.46 

•56 

•65 

•74 

•83 

•3 

•4 

.12 

•25 

•37 

•49 

.62 

•74 

.86 

•99 

i.  ii 

•  4 

•5 

•IS 

•31 

.46 

.62 

•77 

•93 

i.  08 

1.23 

i-39 

•5 

.6 

.19 

•37 

.56 

•74 

•93 

i.  ii 

1.30 

1.48 

1.67 

.6 

•7 

.22 

•43 

-65 

.86 

i.  08 

1.30 

J-73 

1.94 

•7 

.8 

•25 

•49 

•74 

•99 

1.23 

1.48 

1.73 

1.98 

2.22 

.8 

•9 

I.O 

.28 

•56 
.62 

•83 
•93 

i.  ii 
1.23 

i-54 

1.67 
1.85 

1.94 
2.16 

2.22 
2.47 

2.50 
2.78 

•9 

I.O 

.1 

•34 

.68 

1.02 

1.36 

1.70 

2.04 

2.38 

2.72 

3.06 

.1 

.2 

•37 

•74 

I.  II 

1.48 

1.85 

2.22 

2-59 

2.96 

3-33 

.2 

•  3 

.40 

.80 

1.20 

i.  60 

2.OI 

2.41 

2.81 

3-21 

3.61 

•3 

•4 

•43 

.86 

I.30 

1.73 

2.l6 

2-59 

3.02 

3.46 

3-89 

•4 

•5 

.46 

•93 

1.85 

2.31 

2.78 

3-24 

3-7° 

4.17 

•5 

.6 

•49 

•99 

I.48 

1.98 

2.47 

2.96 

3.46 

3-95 

4-44 

.6 

•7 

•05 

2.IO 

2.62 

3-67 

4.20 

4-72 

•7 

.8 

56 

.11 

1.67 

2.22 

2.78 

3-33 

3-89 

4-44 

S-oo 

.8 

•9 

2.0 

:S 

•17 
•23 

I.76 
1.85 

2-35 
2.47 

2-93 
3-09 

3-52 
3-70 

4.10 
4-32 

4.69 
-  4-94 

5-28 

•9 

2.0 

.1 

•65 

•30 

1-94 

2.59 

3-24 

3-89 

4-54 

5.I9 

5-83 

.1 

.2 

.68 

•36 

2.04 

2.72 

3-4° 

4.07 

4-75 

5-43 

6.  ii 

.2 

•3 
•4 

•74 

.42 
.48 

2.13 
2.22 

2.84 
2.96 

3-55 
3-7° 

4.26 
4-44 

4-97 
5-19 

5-68 
5-93 

$$ 

•3 

•4 

•5 

•77 

•54 

2.3I 

3-°9 

3-86 

4-63 

5-40 

6.17 

6.94 

•5 

.6 

.80 

.60 

2.41 

3.21 

4.01 

4.81 

.   5-62 

6.42 

7.22 

.6 

•7 

•83 

.67 

2.50 

3-33 

4-17 

5-00 

5.83 

6.67 

7-5° 

•7 

.8 

.86 

•73 

2-59 

3.46 

4-32 

5-19 

6.05 

6.91 

7.78 

.8 

•9 

.90 

•79 

2.69 

3.58 

4.48 

5-37 

6.27 

7.16 

8.06 

•9 

3-0 

•93 

•85 

2.78 

3-7° 

4-63 

5.56 

6.48 

7.41 

8-33 

3.0 

.1 

.96 

.91 

2.87 

3-83 

4.78 

5-74 

6.70 

7-65 

8.61 

.1 

.2 

•99 

.98 

2.96 

3-95 

4-94 

5-93 

6.91 

7.90 

8.89 

.2 

•3 

.02 

.04 

3-06 

4.07 

5-09 

7-13 

8.15 

9.17 

•3 

•4 

•5 

a 

.10 

.16 

3-15 
3-24 

4.20 
4-32 

5-25 

5-4° 

6.30 
6.48 

7-35 
7.56 

8.40 
8.64 

9-44 
9.72 

•4 
•5 

.6 

.11 

.22 

3-33 

4.44 

5.56 

6.67 

7.78 

8.89 

10.00 

.6 

;j 

.14 

.28 

•35 

3-43 
S-S2 

4-57 
4.69 

5-71 
5-86 

6.85 
7.04 

7-99 

8.21 

9.14 
9-38 

10.28 
10.56 

i 

•9 

.20 

.41 

3-6i 

4.81 

6.  02 

7.22 

8-43 

9.63 

10.83 

•9 

4.0 

•23 

•47 

3-70 

4-94 

6.17 

7.41 

8.64 

9.88 

II.  II 

4.0 

.1 

.2 

.27 

•3° 

•53 
•59 

3-80 
3-89 

5-06 
5-19 

6.33 
6.48 

7-59 
7.78 

8.86 
9.07 

10.12 

IO-37 

"•39 
11.67 

.1 

.2 

•3 

•33 

•65 

3.98 

5-31 

6.64 

7.96 

9.29 

10.62 

11.94 

•3 

•4 

•36 

.72 

4.07 

5-43 

6.79 

8.15 

9-Si 

10.86 

12.22 

•4 

•5 

•39 

.78 

4.17 

5.56 

6.94 

8-33 

9.72 

n.  ii 

12.50 

•5 

.6 

.42 

2.84 

4.26 

5-68 

7.10 

8.52 

9.94 

11.36 

12.78 

.6 

•7 

•45 

2.90 

4-35 

5-80 

7-25 

8.70 

10.15 

1  1.  60 

13.06 

•7 

.8 

•9 

.48 

2.96 
3.02 

4-44 
4-54 

5-93 
6.05 

7.41 

7-56 

8.89 
9.07 

iQ-37 
10.59 

".85 

12.  IO 

13-33 
I3.6l 

.8 
•9 

5-0 

•54 

3-°9 

4.63 

6.17 

7.72 

9.26 

10.80 

12-35 

13.89 

5-° 

co-c, 

1 

2 

3 

4 

5 

6 

7 

8 

9 

eo-c, 

Tables. 


197 


Table  of  Prismoidal  Corrections. 


%-* 

1 

a 

3 

'4 

5 

6 

• 

8 

9 

co-c, 

/w>, 

1-57 

3-15 

4-72 

6.30 

7.87 

9.44 

11.02 

12.59 

14.17 

5-i  * 

.2 

i.  60 

3-21 

4.81 

6.42 

8.02 

".23 

12.84 

14.44 

.2 

•3 

1.64 

3-27 

4.91 

6-54 

8.18 

9.'8? 

"•45 

13.09 

14.72 

•3 

•4 

1.67 

3-33 

5-00 

6.67 

8.33 

IO.OO 

11.67 

J3-33 

15.00 

-4 

•5  • 

1.70 

3-40 

5-09 

6.79 

8.49 

10.19 

11.88 

13-58 

15.28 

-5 

.6 

I-73 

3.46 

5.19 

6.91 

8.64 

iQ-37 

12.10 

13.83 

15-56 

.6 

•7 

1.76 

.3-52 

5-28 

7.04 

8.80 

10.56 

12.31 

14.07 

15-83 

-7 

.8 

1.79 

3-58 

5-37 

7.16 

8-95 

10.74 

12-53 

14-32 

16.11 

.8 

•9 

1.82 

3-64 

5.46 

7.28 

9.10 

10.93 

12-75 

14-57 

16.39 

•9 

6.0 

1-85 

3-7° 

5.56 

7.41 

9.26 

n.  ii 

12.96 

14.81 

16.67 

6.0 

.1 

1.88 

3-77 

5.65 

7-53 

9.41 

11.30 

I3.l8 

15.06 

16.94 

.1 

.2 

•3 

1.91 
1.94 

3-83 

5-74 
5-83 

7-65 
7.78 

9-57 
9.72 

11.48 
11.67 

13.40 
13.61 

15-31 
15-56 

17.22 

.2 
•3 

•4 

1.98 

3-95 

5-93 

7.90 

9.88 

11.85 

I3.83 

15.80 

17-78 

•4 

•5 

2.01 

4.01 

6.  02 

8.02 

10.03 

12.04 

14.04 

16.05 

18.06 

-5 

.6 

2.04 

4.07 

6.  1  1 

8.15 

10.19 

12.22 

14.26 

16.30 

18.33 

.6 

•7 

2.07 

4.14 

6.  20 

8.27 

10.34 

12.41 

14.48 

16.54 

18.61 

•7 

.8 

2.10 

4.20 

6.30 

8.40 

10.49 

12.59 

14.69 

16.79 

18.89 

.8 

•9 

2.13 

4.26 

6-39 

8.52 

10.65 

I2.78 

14.91 

17.04 

19.17 

•9 

7-0 

2.l6 

4-32 

6.48 

8.64 

10.80 

12.96 

15-12 

17.28 

19.44 

7.0 

.1 

2.19 

4.38 

6-57 

8.77 

10.96 

I3.15 

15-34 

17-53 

19.72 

., 

.2 

2.22 

4-44 

6.67 

8.89 

ii.  ii 

I3.33 

17.78 

20.00 

.1 

•3 

2.25 

6.76 

9.01 

11.27 

13.52 

15-77 

18.02 

20.28 

'     -3 

•4 

2.28 

4-57 

6.85 

9.14 

11.42 

I5.99 

18.27 

20.56 

•  4 

•5 

2.31 

4-63 

6-94 

9.26 

"•57 

I3.89 

16.20 

18.52 

20.83 

-5 

.6 

2-35 

4.69 

7.04 

9-38 

"•73 

14.07 

16.42 

18.77 

21.  II 

.6 

•7 
.8 

2.38 
2.41 

4-75 
4.81 

7.22 

9-'S 

11.88 
12.04 

14.26 
14.44 

16.64 
16.85 

19.01 
19.26 

21.39 
21.67 

.1 

•9 

2.44 

4.88 

7-3i 

9-75 

12.19 

I4.63 

17.07 

19.51 

21.94 

•9 

8.0 

2-47 

4.94 

7.41 

9.88 

12.35 

I4.8l 

17.28 

19-75 

22.22 

8.0 

.1 

2.50 

5-oo 

7-50 

10.00 

12.50 

I5.OO 

17-5° 

20.00 

22.50 

.1 

.2 

2.53 

5-o6 

IO.I2 

12.65 

15.19 

17.72 

20.25 

22.78 

.2 

•3 

2.56 

5.12 

7.69 

10.25 

12.81 

!5-37 

20.49 

23.06 

•3 

•4 

2.59 

5.19 

7.78 

IO-37 

12.96 

I5-56 

18.15 

20.74 

23-33 

-4 

•5 

2.62 

5-25 

7-87 

10.49 

13.12 

15-74 

18.36 

20.99 

23.61 

•5 

.6 

2.65 

5-31 

7.96 

10.62 

13-27 

15-93 

18.58 

21.23 

23.89 

.6 

•7 

2.69 

5-37 

8.06 

10.74 

J3-43 

16.  ii 

18.80 

21.48 

24.17 

•7 

.8 

2.72 

5-43 

8.15 

10.86 

13-58 

16.30 

19.01 

21-73 

24.44 

.8 

•9 

2-75 

5-49 

8.24 

10.99 

13-73 

16.48 

19.23 

21.97 

24.72 

•9 

9.0 

2.78 

5.56 

8-33 

ii.  ii 

13.89 

16.67 

19.44 

22.22 

25.00 

9.0 

.1 

2.81 

5-62 

8.43 

11.23 

14.04 

16.85 

19.66 

22.47 

25.28 

.! 

.2 

2.84 

5-68 

8.52 

11.36 

14.20 

*7-f4 

19.88 

22.72 

25  56 

.2 

•3 

2.87 

5-74 

8.61 

11.48 

14-35 

17.22 

20.09 

22.96 

25.83 

-3 

•4 

2.90 

5-8o 

8.70 

ii.  60 

17.41 

20.31 

23.21 

26.11 

•4 

•5 

2-93 

5-86 

8.80 

"•73 

14.66 

17-59 

20.52 

23.46 

26.39 

•5 

.6 

2.96 

5-93 

8.89 

".85 

14.81 

17.78 

20.74 

23.70 

26.67 

.6 

•2 

2.99 

5-99 

8.98 

11.98 

14.97 

17.96 

20.96 

23-95 

26.94 

•7 

.0 

3-02 

6.05 

9.07 

12.  IO 

15-12 

18.15 

21.17 

24.20 

27.22 

.8 

•9 

IO.O 

3.06 
3-09 

6.V7 

9.17 
9.26 

12.22 
12-35 

15.28 
15-43 

18.33 
18.52 

21.39 
21.60 

24.44 
24.69 

27.50 
27.78 

•9 

IO.O 

«v-* 

1 

* 

3 

4 

5 

6 

^ 

8 

9 

«,-«, 

198         Railroad  Curves  and  Earthivork. 


ALLEN'S  TABLES  (Copyright,  1893,  by  C.  F.  ALLEN). 
Triangular  Prisms.    S  in  cu.  yds.  for  50  ft.  in  length. 


WIDTH 

1 

» 

3 

4 

5 

6 

^ 

8 

9 

WIDTH 

Height 

0.1 

.09 

.19 

.28 

•37 

.46 

.56 

•65 

•74 

•83 

Height 

O.I 

.2 

.19 

•37 

.56 

•74 

•93 

i.  ii 

1.30 

1.48 

1.67 

.2 

•3 

.28 

•56 

•83 

i.  ii 

1-39 

'1.67 

1.94 

2.22 

2.50 

•3 

•4 

•37 

•74 

i.  ii 

1.48 

1.85 

2.22 

2.59 

2.96 

3-33 

•4 

•5 

.46 

•93 

1.39 

1.85 

2.31 

2.78 

3-24 

3-70 

•5 

.6 

.56 

i.  ii 

1.67 

2.22 

2.78 

3-33 

3.89 

4-44 

5.00 

.6 

•7 

•65 

1.30 

1.94 

2-59 

3-24 

3-89 

4-54 

5-19 

5.83 

•7 

.8 

•74 

1.48 

2.22 

2.96 

3-70 

4-44 

5-19 

5-93 

6.67 

.8 

•9 

•83 

1.67 

2.50 

3-33 

5-oo 

5-83 

6.67 

7-5° 

•9 

x.o 

•93 

1.85 

2.78 

3-70 

til 

5.56 

6.48 

7.41 

8-33 

I.O 

.1 

1.02 

2.04 

3.06 

4.07 

5-09 

6.ii 

7-I3 

8.15 

9.17 

j 

.2 

i.  ii 

2.22 

3-33 

4-44 

5.56 

6.67 

7.78 

8.89 

IO.OO 

.2 

•3 

1.20 

2.41 

3-61 

4.81 

6.  02 

7.22 

8.43 

9.63 

10.83 

.3 

•4 

1.30 

2-59 

3.89 

5-19 

6.48 

7.78 

9.07 

IO-37 

11.67 

.4 

•5 

x-39 

2.78 

4.17 

5.56 

6.94 

8-33 

9-72 

ii.  ii 

12.50 

•5 

.6 

1.48 

2.96 

4-44 

5-93 

7.41 

8.89 

IO-37 

11.85 

13-33 

.6 

•7 

1.57 

3-15 

4.72 

6.30 

7.87 

9-44 

11.02 

12.59 

14.17 

•7 

.8 

1.67 

3-33 

5-00 

6.67 

8-33 

10.00 

11.67 

J3-33 

15.00 

.8 

•9 

1.76 

3-52 

5.28 

7.04 

8.80 

10.56 

12.31 

14.07 

15-83 

•9 

2.0 

1.85 

3-70 

5.56 

9.26 

ii.  ii 

12.96 

14.81. 

16.67 

2.0 

.1 

.2 

1.94 
2.04 

3.89 
4.07 

5.83 

7.78 
8.15 

9.72 
10.19 

11.67 

12.22 

I3.6l 
I4.26 

i|,5« 

16.30 

I7-5° 

18.33 

.1 

.2 

•3 

2.13 

4.26 

6-39 

8.52 

10.65 

12.78 

14.91 

17.04 

19.17 

•3 

•4 

2.22 

4-44 

6.67 

8.89 

ii.  ii 

I3-33 

'5-56 

17.78 

20.00 

.4 

•5 

2.31 

4-63 

6-94 

9.26 

"•57 

13.89 

16.20 

18.52 

20.83 

•5 

.6 

2.41 

4.81 

7.22 

9-63 

12.04 

14.44 

16.85 

19.26 

21.67 

.6 

•7 

2.50 

5.00 

7-5° 

IO.OO 

12.50 

15.00 

17.50 

20.00 

22.50 

•7 

.8 

2-59 

5-19 

7.78 

10.37 

12.96 

15.56     18.15 

20.74 

23-33 

.8 

•9 

2.69 

5-37 

8.06 

10.74 

I3-43 

i6.n     18.80 

21.48 

24.17 

•9 

3-o 

2.78 

5-56 

8-33 

II.  II 

13.89 

16.67 

19.44 

22.22 

25.OO 

3-0 

.1 

2.87 

5-74 

8.61 

11.48 

I4-35 

17.22 

20.09 

22.96 

25.83 

.1 

.2 

2.96 

5-93 

8.89 

11.85 

14.81 

17.78     20.74 

23.70 

26^7 

.2 

•3 

3-06 

6.ii 

9.17 

12.22 

15.28 

18.33  :  21.39 

24.44 

27.50 

•3 

•4 

3-15 

6.30 

9.44 

12.59 

J5-74 

18.89  22.04 

25.19 

28.33 

•4 

•5 

3-24 

6.48 

9.72 

12.96 

16.20 

19.44  22.69 

25-93 

29.17 

•5 

.6 
•7 

3-33 

3-43 

6.67 
6.85 

10.00 

10.28 

x3-33 
13-70 

16.67 

20.00 
20.56 

23.33 
23.98 

26.67 
27.41 

30.00 
30.83 

.6 
•7 

.8 

3-52 

7.04 

10.56 

14.07 

17-59 

21.  II 

24.63 

28.15 

3L67 

.8 

•9 

7.22 

10.83 

14.44 

18.06 

21.67 

25.28 

28.89 

32.50 

•9 

4.0 

3-70 

7.41 

II.  II 

14.81 

18.52 

22.22 

25.93 

29.63 

33-33 

4.0 

.1 

3-80 

7-59 

"•39 

I5<Ig 

18.98 

22.78 

26.57 

30-37 

34.17 

.1 

.2 

3-89 

7-78 

11.67 

15.50 

19.44 

23-33 

27.22 

3I.II 

35.00 

.2 

•3 

3.98 

7.96 

11.94     15.93 

19.91 

23.89 

27.87 

3I-85 

35-83 

•3 

•4 

4.07 

8.15 

12.22       16.30 

20.37 

24.44 

28.52 

32.59 

36.67 

•4 

•5 

4.17 

8-33 

I2.5O 

16.67 

20.83 

25.00 

29.17 

33-33 

37-50 

•5 

.6 

4.26 
4-35 

8.52 
8.70 

12.78       17.04 
13.06       17.41 

21.30 
21.76 

25.56 
26.11 

29.81 
30.46 

34-07 
34.81 

38.33 
39-17 

.6 

.8 

4.44 

8.89 

J3-33     I7-78 

22.22 

26.67     31-" 

35-56 

40.00 

.8 

•9 

4-54 

9.07 

13.61     18.15 

22.69 

27.22     31.76 

36.30 

40.83          .9 

5-0 

4-63 

9.26 

13.89     18.52 

23.I5 

27.78     32.41 

37-04 

41.67 

5-0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diagram 

for 

THREE  LEVEL  SECTIONS 
Base  14.     Slope  1%  to  1 


Center  Heights  on  Oblique  Lines 

Sum  of  Distances  Out  on  Vertical  Lines 

Quantities  on  Horizontal  Lines  in 

cubic  yards  for  50  ft.  of  Length 


Diagram 

for 

THREE  LEVEL  SECTIONS 
Base  20.     Slope  iy2  to  1 


Quantities  on  Horizontal  Lines  in  cubic 
yards  for  60  ft.  of  Length 


Diagram 

for 
PRISMOIDAL  CORRECTION 


Differences  between  Sum  of  Distances 
out  on  Vertical  Lines 
Differences  between  Center  Heights  on 
Oblique  Lines 


Quantities  on  Horizontal  Lines  In  cu. 
yds.  for  100  ft.  of  Length 


Diagram 

for 
TRIANGULAR  PRISMS 


Base  on  Vertical  Lines 
Altitude  on  Oblique  Lines 
Quantities  on  Horizontal  Lines  In 
cubic  yards  for  50  ft.  of  Length 


50 


4C 


3C 


20 


10 


THIS  15OUJK.  IS  L»U±J   OJN    THIS  .LAST   DAT±i 

STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


LD  21-95m-7,'3'i 


~*Vki'il¥ 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


